Chapter 4 Star Bodies

Chapter 4 Star Bodies

CHAPTER 4 STAR BODIES In this chapter the relations between arbitrary star bodies S and arbitrary lattices A are studied in more detail. The theory ...

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In this chapter the relations between arbitrary star bodies S and arbitrary lattices A are studied in more detail. The theory to be ekposed has for the greater part been developed by Mahler. First, we derive continuity properties of certain functionals depending on S or A. Next, we consider points of S-critical lattices lying on, or near to, the boundary of S. Thereafter, we study some types of reduction of a star body; by this we mean reduction to smaller star bodies having the same critical determinant. In particular, we consider star bodies admitting an infinite group of automorphisms. We shall repeatedly make use of Mahler's selection theorem which says that a bounded sequence of lattices contains a convergent subsequence (sec. 17). 25. The functionals A ( S ) , I'(S), !(A),


25.1. Let S be a star body, with distance functioiif. We recall that the critical determinant d ( S ) is given by

d(S) = inf {d(A): A admissible for S } , with the understanding that d(S)= 00 if there are no S-admissible lattices. The star body S is said to be of thefinite or of the infinite type according to whether d(S) is finite or infinite. A bounded star body is of the finite type; an unbounded star body may be of the finite or of the infinite type. The quantity d(S) is always > 0, as S contains a neighbourhood of 0 . If S is of the finite type, then according to theorem 17.9 there exist S-critical lattices, i.e., lattices which are admissible for S and have determinant d(S). In the sequel, we shall often make use of this fact without mentioning it explicitly. It should be observed that the critical determinant of a star set may be zero. Thus the lemniscate {(x1)2+(x2)2)2

s (xi)2-(x2)2

o 25



29 1

and, more generally, each bounded star set A4 for which there is a straight line L through 0, such that M n L = {o}, has critical determinant 0 (Mahler [25a], Rogers [27b]). In this chapter we denote by So the star body given by





It is unbounded, as it contains the hyperplanes xi = 0. But it is of the finite type, because there exist &-admissible lattices (theorem 4.1). We shall see that in many cases it is the simplest example to illustrate the general theory. 25.2. We begin by studying the behaviour of the quantity A ( S ) . Let G be the collection of star bodies of the finite type. In G, one can introduce

a topology in various different ways. But only if a concept of limit is defined in a rather special way, one arrives at a property of the form (2)

A(T) + d(S)

if T


( S , T E (5).

For arbitrary star bodies S, T we might define (31

6(S, T ) = V(S\T)+ V(T\S).

One can easily show that 6 defines a metric in G (compare sec. 1.5; observe, however, that 6(S, T ) may be infinite if S or T is unbounded). But d(S) is not even continuous, in this metric, on the collection of bounded star bodies. This is illustrated by the following example (Rogers [26a]). Let S be the square lxil =< 1 ( i = 1 , 2 ) and let S, (0 < 9 < 1) be the part of S given by

.51x21 5 1,


Then S, is a star body, and 6(S, S,) A ( S ) = 1, A&) = 4 (0 < 9 < 1). It is true that

1x11 6 1.

= $9 + 0

as 9 + 0. But we have

lim sup A(&) 6 A ( S ) ,+O-

if S, S, are star bodies which are uniformly bounded and 6(S, S,)- 0 for 9 + 0 (Rogers [26a]). However, as we shall see in the next section, even the weaK relation (4)ceases to be generally true if the first condition is not fulfilled.





In spite of this, Mahler [25c] could prove a result of the form (2). Let S") = S n C,, C, denoting the sphere 1x1 _I 2 ( 2 > 0). For E, 't > 0, call the ( E , 2)-neighbourhood of S the collection of all T E G such that

T c (l+&)S, s'" c ( l + E ) T .


Thus, loosely speaking, a neighbourhood of S E G is the collection of all T E G which, in a specified sense, are approximately enclosed between S(r)and S. The system of all ( 8 , z)-neighbmrhoods of all SEG defines a topology on S which may be called the ( E , 2)-topology*. It is with respect to this topology that (2) is true. One has

Theorem 1. Let S, S, E (5 (r = 1,2, . . .). Then limd(S,) = d(S) ?+W

ift in the ( E , T)-topology, S, tends

to S f o r r .+ GO, i.e.,

q,f o r suficiently

large r, S, is contained in any given ( E , 2)-neighbourhood of S. In particular, lim A(s(')) = A(s).



Proof. First we prove the relation (7). Clearly, for each r, d(S"') 5 d(S). Suppose that, for some sequence of indices ri --f a, (8)

lim d(S(I0) exists and is less than d(S).


For i = 1,2, . . ., let A,, be an S("'-critical lattice. Then d(A,,) is bounded. Moreover, A,, is admissible for a fixed neighbourhood of 0,e.g., for S(l). Hence, by the selection theorem, the sequence (Ari)contains a convergent subsequence. We may suppose that the indices ri are so chosen that the sequence {A,,} itself converges to some lattice A , say. We show that this assumption leads to a contradiction. We have d(A) = lim d(A,,) = lim A(S(,'))< d(S). Hence, A has a point # o in the interior of S. So it has a point in the interior of some S'". From the assumption that A,, converges to A for i + GO it follows then that A,, has a point # o in the interior of S"), provided i is sufficiently large. Since ri > z for sufficiently large i, this contradicts the choice of the

* With this topology, (5 becomes a Hausdorff space in which the first axiom of countability holds. The subsequent theorem 1 is equivalent with saying that, in this space, d (5') is continuous. Note that the (E, 7)-neighbourhoodsare not open.

I 25




lattices A,<. Thus there is no sequence of indices r i + co for which (8) holds. This proves (7). Now let E, t be two positive numbers. Then (5) holds, with T = S,, if r is sufficiently large. Hence, d(S,) S (l+~)”d(S)and A(S“))

S (l+~)”d(S,),

if r is sufficiently large. On account of (7), this proves (6). We remark that the relation (7) was used earlier by Mahler [25b] as a definition of d(S). The relations ( 6 ) and (7) are not generally valid for star sets, as is seen by considering the two-dimensional set 0 5 x1x2 S 1 (Mahler [25a]). Mahler [25d] also deduced, under less stringent conditions, the following weaker result.

Theorem 2. Let S, S, be star bodies in 6,with distance functions f, f,, respectively ( r = 1,2, . . ,). Suppose thatf,(x) +f (x) as r -+ co, uniformly on the sphere 1x1 = 1. Then lim inf d(S,) 2 d(S).




Indeed, it follows from the conditions of the theorem that 3 (I - E ) S ( ~ )for , any given E, z, if r is sufficiently large.

Mahler illustrated by the following example that in (9) the inequality sign may hold. Put S, = So u T,, where T, is given by r-’((xl)’

+ - - +(xn- J’) + r2(n-1)(xn)2s a2 *

and where a > 0 is so chosen that Tl = C, has critical determinant d(Tl) > d(So). On account of the relations (17.11) and (17.13), A(&) 2 d(T,) = d(T,) for all r. Hence, (9) does not hold with the equality sign. Nevertheless, it can be proved that the star bodies S, satisfy the conditions of theorem 2. 25.3. For bounded convex bodies the situation is much simpler. We use the metric 6 given by (3). Let KO,K be bounded convex bodies containing o as an inner point, KObeing fixed. Then, for any E > 0,

(l+e)-lK, c K c (l+&)KO


CH. 4


if 6(K, K O ) is sufficiently small (sec. 1.5). Hence, by (17.12), (10)

A ( K ) + A(K,)

if 6 ( K , KO)-+ 0.

In other words, A ( K ) is continuous in the metric S . Macbeath [25a] considered classes of affinely equivalent convex bodies. We shall give a simple proof of his main result. For simplicity, we restrict. ourselves to symmetric bodies, but this restriction is not essential. Let 9 denote the collection of all bounded symmetric convex bodies K in R". Further, let G be the group of the non-singular affine transformations of R".Then we have the following Theorem 3. Let a ( K ) be a real function on R which is conrinuous in the metric 6 and which is invariant under the transformations in G . Then u ( K ) attains a maximum and a minimum value on R. The conditions of the theorem are satisfied for a ( K ) = d ( K , ) / Y ( K ) , where K , has the form K , = K - a and has centre at 0.This follows from (10) and (17.13). Thus we have Corollary. On the collection of bounded o-symmetric convex bodies K, the quotient A ( K ) / V ( K )attains a minimum calue. Proof of theorem3. Put a = inf { a ( K ) :K E R}. Take a sequence of bodies K, E 9 such that a(K,) --f u if r -+ co. By theorem 1.8, there exist ellipsoids E, such that n-+E, c K, c E, (r = 1,2, . . .). Let A , be an affine transformation such that A,E, is the unit sphere C , . Then we have (11)

n - ) C , c A,K, c C ,

( r = 1,2, . . .),

whereas A,K, E R and a(A,K,) = a(K,) a if r -+ 00. To the sequence {A,K,} we apply the selection theorem of Blaschke (sec. 1.5). We find that there is a sequence of indices r, such that the sequence of bodies A,,K,, converges to a convex set K ' . FrcJm (11) it follows that n-*C, c K' c C1,and so K' E 9. Moreover, since a ( K ) is continuous, a ( K ' ) = lim a(A,K,) = a. Hence, a ( K ) attains on 9 the value a. Similarly, it attains the value sup {cc(K): K E R}. This proves the theorem. --f

Macbeath proved the theorem by showing that the classes of affinely equivalent bodies Kconstitute a compact metric space 9*.More precisely,

o 25

THE FUNCTIONALS d(s), r ( s ) , f ( A )g(A) ,


he showed that the following function induces a metric in R*: z(K1, K2) = log P(K1 K,)+log P(K2 Y Kl), Y

where p(K1, K 2 ) = inf(V(AK2)/V(Kl):A


and K l

= AK2).

Mahler (25e, 25f) obtains some special results concerning the determinant of a variable convex body in the plane: lo. Let K be an o-symmetric bounded convex body in RZ and let K, be its intersection with the strip lxzl 6 CY (CY> 0). Then a-'A(K,,) is a nonincreasing function of 0. In other words, if z > oYthen d(K,) 5 (z/cY). .A(&). This result can be obtained by subjecting a K-critical lattice to a transformation leaving invariant the points of the x,-axis and a pair of tac-lines to K at the points of the boundary lying on the x,-axis. See also Woods [18a]. 2". Let K , , Kz be two o-symmetric bounded convex bodies in R2. By using the method of the circumscribed symmetric hexagons (sec. 22.1) and by applying the theorem of Brunn-Minkowski (sec. lS), Mahler finds that

{ ~ ( ~ K , + ( ~ - L J ) K , )2) *9 ( ~ ( ~ , ) } * + ( 1 - 9 ) ( d ( ~ , ) )(0 i < LJ < I). 25.4. For the covering constant T ( M ) studied in sec. 21, there is a result

similar to theorem 1. We consider arbitrary star bodies. Bambah [21a] proves

Theorem 4. Let a sequence S, converge to S in the star bodies). Then one has (12)


T)-topology (S, S,

lim r(s,)= r(s);in particular, lim r(P)= r ( ~ ) .

r-+ 03


Proof. First we prove the last relation (12). Clearly, r(S(')) is nonincreasing and 6 T(S). Hence, y = lim T(S")) exists; moreover, y 5 T(S).We suppose that y < T ( S ) (this implies that y is finite) and deduce a contradiction. Choose E > 0 such that y(1+2e)" < r ( S ) .Then there exists a covering lattice A' of S with determinant d(A') = y(1+2~)". Put A = (1 +&)-'A' and take a fundamental cell P of A. Then A is a covering lattice of (1 +e)-lS, and so to each point z E P one can find a point x E A such that z E (1 + ~ ) - l S f x . The set S+x then contains a certain neighbour-





hood of z . It follows that P is covered by finitely many bodies S+x', xi E A . It is also cavered by the bodies S(r)+xi,if r is sufficiently large.

contraHence, for sufficiently large r, A is a covering lattice of S"), In ' diction with y < d(A) = y(1+2~)"(1+~)-".This proves the theorem in the special case S, = S"'. In the general case, we argue as follows. Let E , T > 0 be arbitrary (z integral). Then, for sufficiently large r, Sr c (1 + E ) S and S") c c (1 &)S,,hence




r (l+&))T(S,)

(see the relations (21.2) and (21.3)). Since limT(S(')) theorem follows.


T ( S ) , the

25.5. Let S be a fixed star bady of the finite type. We wish to study the behaviour of the homogeneous minimum A(S, A), as a function of A.

In terms of the distance function f determined by S, this minimum is the greatest lower bound of f ( x ) , where x runs through the points # o of A (sec. 17.4). Therefore, we denote it byf(A), so that we have

f(A) = A(S, A ) = inf ( f ( x ) : x # o and x E A } . If A is admissible for S, thenf(A) 2 1; if A is S-critical, then, clearly, f ( A ) = 1. This proves the following theorem (Mahler [25c]). Theorem 5. If A is S-critical, therc for each xEAwith1 5 f ( x ) < 1 + ~ .


> 0 there exists a point

The function f ( A ) is (in the case that S is unbounded) decidedly not continuous in the topology which we introduced in the set of lattices (sec. 17.1). Take, e.g., S = S o , and consider an admissible lattice A . There are rational lattices A' which are arbitrarily near to A. Such a lattice Ar is homothetic to a sublattice of Y; therefore, it has points x OA any fixed coordinate axis. These points x satisfy f ( x ) = 0. Hence, f(Ar)= 0 for each rational lattice, whereasf(A) 2 1. Thusf(A) is not continuous. A similar argument holds in the general case. For it is no loss of generality to suppose that S contains one of the coordinate axes. Nevertheless, one can prove weaker assertions. A first result of Mahler [25c] in this direction can be expressed by


8 25

r(s),f(n),S ( n )


Theorem 6. The function f ( A ) is upper semi-continuous, or what comes to the same thing, (13)

lim supf(A,) r-ca

5 f ( A ) f o r each sequence




Proof. Let A, + A and let x # o be any point of A. By theorem 17.1, there is a sequence of points xr + x with xr E A,. For sufficiently large r, we have xr # 0, and so f (x') 2 f ( A r ) . Hence, by the continuity off,

f ( x ) = limf(x') 2 lim supf(A,). Since x and the A, were arbitrary, (13) follows.

Corollary". I f a sequence of S-admissible lattices A, converges to a lattice A, then A is S-admissible. In certain cases, the equality sign holds in (13). This is illustrated by the next theorem.

Theorem 7. Let A, -+ A and let there exist two positive constant3 a, p and a sequence of points xr with the following properties 1". xr # 0 , Y EA,, lx'l 5 p (r = 1 , 2 , . . .) 2". f ( x ' ) + a and f ( A , ) + u as r + co.

Then f ( A ) = a. Moreover, A contains a point x w i t h f ( x ) = a. Proof. By definition, there exist bases A , A, of A, A,, respectively, such that A, + A if r + co. There further exists a positive number y such that (14)

[A,yl 2 y

if lyl = 1 and r is sufficiently large.

NQWlet xr = A , d (r = 1 , 2 , . . .). Then, by 1" and (14), Id1 5 B y - ' . It follows that there is an infinite sequence of indices rl , r 2 , . . ., such that u" = u is constant. Clearly, x" + Au if r + 00, Therefore, x = lim x" exists, and this point belongs to A. Moreover, f ( x ) = = lim f (xrr)=a. Next, f(A,,) + a. Hence, by theorem 6, f ( A ) 2 a. Since x E A and f ( x ) = a, we also have f ( A ) a. This completes the proof of the theorem.


See the proof of theorem 17.9.



CH. 4

25.6. We introduce the function

g ( A ) = f(A)d(A)-"". This function is constant on each set of homothetical lattices. By theorem 6 and the continuity of d(A), it is upper semi-continuous. Furthermore, it attains a maximum value. Actually, by (17.18) or (17.19), this maximum value is given by max g(A) = A(s)-""


and it is attained for the lattices A which are homothetic to an S-critical lattice. It is quite natural to consider local maxima, too. Cohn [25a] gives the following

Definition 1. A lattice A, is called stable (with respect to the star body S or the distance function f)if g(A,) is a local maximum, i.e., i f , in the space of lattices, there exists a neighbourhood N of A, such that s(A) g(A0) for A E N .


Clearly, if A, is stable, then so is each lattice A which is homothetic to A , . Usually, instead of stable lattices, one employs so-called extremal lattices. The last concept is defined as follows.

Definition2. A lattice A, is called extremal for the star body S i f f (A,) = 1 and, moreover, there exists a neighbourhood N of A, such that (17)

d(A) 2 d(Ao)

if A E N and f ( A ) 2 1.

We note that there need not exist a lattice A # A , with A E N , f ( A ) 2 1. In many cases, such an isolation property of a lattice A , holds in an even stronger sense (see sec. 43). Nevertheless, we can show

Theorem 8. A lattice A, is extremal for S if and only i f f (A,) = 1 and A, is stable. Proof. I f f ( A , ) = 1 and g ( A ) 5 g ( A , ) for all lattices A in some neighbourhood N of A,, then clearly (17) holds for this neighbourhood.

Thus the conditions are sufficient. Now suppose thatf(A,) = 1 and that (17) holds for some N. There

exist a positive number


and a neighbourhood N , of A , such that

1". p A E N i f A E N , and l - E < p < l + E 2". (1-E)" < d(A,)/d(A) < (1+~)nfor A E N , . From 1" and (17) we conclude that (17')

d(A) > d(A,)

if A


N , and f ( A ) > 1.

It is no loss of generality to suppose that (17') holds with N 1 replaced by N . Now take an arbitrary lattice A E N , . Put A' = PA, where p" = = d(A,)/d(A).Then d(A') = d(A,) and A' E N , on account of 1" and 2". Applying (17') we find that, Iiecessarily,f(A') 5 1;this conclusion is also valid in the case that the set of lattices A E N, withf(A) > 1 is empty. Me now have g ( A ) = g(A') 6 d(A')'-''" = d(A,)-l'" = g(A,)

if A

E N1.

This proves that A , is stable and completes the proof of the theorem. To check whether a given lattice A , is stable with respect to a given distance function f is, to a certain extent, a problem in calculus. Cohn [25a] treats this problem in the following way. Suppose that f ( x ) has the form* f(x) = \tp(x)\, where q ( x ) is continuously differentiable with respect to the variables x , , . . ., x,, . For an arbitrary lattice A = AY and for u E Y,put $(u, A) = q(Au)(det A)-"".

Then g ( A ) = inf(llC/(u,A ) ] :u # o and U B Y } . We wish to study the variations of $(u, A ) caused by small variations of A = (aij); to this end we introduce the n2-vector o(u, A ) with components




o i j ( u , A ) = [email protected](u,A ) = -{p(Au)(det A)-"") aaii aaij

( i , j = 1,.

. ., n).

Generally speaking, the vectors o(u, A ) , u E Y generate a linear subspace of dimension q < n2. The number q is called the free dimension of A with respect to the function f.One can easily show that q is equal to the dimension of the linear subspace of R"' generated by the vectors

* In applications it may be convenient to work with F(x) = ]p(x)l", K a constant

> 0. In (19), one has then to replace 1/n by K / n .


cn. 4


~ ( x )x ,E R", with components 1 Xij(x) = a q ( x ) x j - -q(x)8ij n


(i,j = 1 , . .,n),

6 , being the Kronecker delta. In particular, q does not depend on A. In the casef(x) = l x l x z . * xnl'"' one has q = n(n- 1). Now suppose we have a number of vectors d' E Y with $(d,A , ) = = g(A,), A, = A , Y. Let A = A , + B , where the elements of B = (b,) are small. We have $(Uk,

A)-$(Uk, A,) =


(0 < s k < 1).

i, j

If, for each choice of the b i j , we can find an index k, such that the last expression is 5 0, then g ( A ) attains a maximum at A , . In this way we are led to the following

Theorem 9. Let f ( x ) = Iq(x)l, where q ( x ) is continuously differentiable with respect to the coordinates x i . Let f ( A )and o ( u , A ) be defined by (12) and (18), respectively, and let q be the free dimension of a lattice with respect to f. Then a lattice A, = A , Y is stable, i f f (A,) > 0 and if there are q f 1 points u l , . . ., u4" E Y wch that 1". f ( A 0 u k )=f(A,) for k = 1 , . . ., q + l 2". the q+ 1 vectors w(u", A ) are positively dependent; by this we mean that these vectors do not lie in a ( q - 1)-dimensionallinear subspace of Rn2 and that there subsists a relation l a k w ( d ' , A ) = 0, where a l , . . ., ctq+ areall > 0.

26. Points of critical lattices on the boundary. Automorphic star bodies 26.1. Let S be a star body. Mahler [25c] treats in detail the following

question, in particular in the case of so-called automorphic star bodies: how many points of a given critical lattice A lie on, or near to, the boundary of S? In the first part of this section we shall treat this question for the more general class of extremal lattices. A first result is given by

Theorem 1. If S is a bounded star body, then each extremal lattice A has n independent points on the boundary of S.

Proof. Let M be the linear subspace generated by the points of A on the

8 26



boundary of M , and let m = dim M. Suppose that m < n. By theorem 3.3, A has a basis (a', . . ., d},such that a' E M for i = 1,. . ., m. Since S is bounded and the points of AM \ do not lie on the boundary of S, there exists a number 9 with 0 < 9 < 1 such that the lattice A' with a basis {a', . . ., a"', , . . ., 9d>is admissible for S. Thus A is not extremal. This proves the theorem. Since S is o-symmetric, it follows from theorem 1 that at least n pairs of points + x E A lie on the boundary of S. In general, but not always, there are more such pairs (for convex bodies, see sec. 26.2). We give the following Definition 1. Atz extremal lattice A of a bounded star body S is called singular i f it has exactly n pairs ofpoints fx on the boundary of S ; if there arc more, then A is called regular. In particular, we distinguish singular and regular critical lattices. Mahler [29b, 25c] gives an example of a bounded, n-dimensional star body S with just one critical lattice which, moreover, is singular. (It is obtained by making small conic incisions in the cube W : lxil 5 (+)'"', in such a way that the points fe' lie on R, and the remaining points u # o of Y n W outside, but near to R,where R is the boundary of the resulting star body S). By an appropriate extension of S one gets a star body S1 with just one critical lattice, such that just m pairs of points $.x of this lattice lie on the boundary of S , , m being any prescribed integer 2 n (compare Hejtmanek [26a]).

26.2. We intend to show that a convex star body K only possesses regular extremal lattices. In the case n = 2, at least for critical lattices, this follows from lemma 22.2 which says that, in this case, a critical lattice has at least 3 pairs of points fx on the boundary of K. In the case n = 3, one can show that there are at least 6 such pairs (we shall come back to this question in sec. 31). Swinnerton-Dyer [26a] generalizes this result in two directions by proving

Theorem 2. Let K be a bounded o-symmetric convex body in R". Then each extremal lattice A of K has at least +n(n+ 1) pairs of points + x on the boundary of K.


CH. 4


Proof. We may suppose that A = Y. Let ILL',. . ., +u" be the points of Yon the boundary of K. Further, let A' be a neighbouring lattice of the form A' = (I+qA)Y, where I is the unit matrix, A = ( a i j ) is a matrix to be chosen below and 1q1 is small. Suppose that m < +n(n+ 1). We show that then, for suitable choices of A and q, A' is K-admissible and has determinant d(A') < 1 = d(A). Put xk = ( I + qA)uk (k = 1, . . ., n). We can arrange that xk belongs to any fixed tac-plane ( x - u k ) . ck = 0 to K at the point uk, for k = 1, . . ., m, by choosing the elements aij of the matrix A so as to satisfy the equations ( A u k ) .C k =


UijUjkCik =

0 (k


1 , . . ., rn)

i. j

It is even possible to choose the aij, not all zero, such that the m relations (1) hold and, in addition, aij=ujifor i#j. For we have m+$n(n- l ) < n 2 . We now observe that the points xk do not belong to the interior of K. It is then clear that A' is K-admissible if 1 ~ is1 sufficiently small. Now consider d(A'). We have

d(A') = det(I+qA) = 1+p,q+pzq2+

* . -


where i

Hence, 2p,-p:

= -2

c aijaji- c u;

i< j



c {(aij-uji)z-(uij+aji)2}- c

i< j



Since a i j = aji (i # j ) , we may conclude that 2p, - 0: < 0. Hence, either fil # 0 or PI = 0, b2 < 0. In both cases, it follows that d(A') < 1, if Iql is small enough and q has the appropriate sign. We conclude that A' is not extremal. So we have m 2 +n(n+l). This completes the proof. On the other hand, an arbitrary K-admissible lattice A has at most 3"-1 points on the boundary of K . To see this suppose that A = Y. Let u, u be two distinct points of Y on the boundary of K. Then +(u- u ) is a point # o in the interior of K and therefore not a lattice point. Consequently, u # u (mod 3). Since we cannot have u 5 o (mod 3), the assertion follows.

§ 26



The italicized assertion holds with 3"-l replaced by 2"- 1, in the case that K is strictly convex (confer sec. 9.4). It was generalized to the case of a bounded star body S with a concavity coefficient ws by Groemer [26a].

26.3. For unbounded star bodies the situation is quite different. On the one hand, theorem 1 need not remain true; on the other hand, A may have an infinity of points on the boundary of S (sec. 26.4). We confine ourselves to critical lattices. Theorem 25.5 is a first result of the type desired. A slight refinement is given by Theorem 3. Let S be an (unbounded) star body, with distance function$ Suppose that A ( S ) < 00. Then,for euch E > 0,there exists a number z > 0 such that each S-critical lattice A contains a point x with 1 5 f (x)< 1 + E, 1x1

6 2.

Proof. By (25.7), there exists a pxitive integer z with A((1 +$&)S(*))> > A ( S ) . Each S-critical lattice A contains a point x E (1 +&)S('), x # 0 ; this point satisfies the assertions. By the following example, Mahler [25c] shows that there may be no point x E A with f ( x ) = 1. Consider So, with distance functionfo(x) = = 1x1 x2 * X"l i'n, and let S be the star body with distance function a


Iff(x) = 1,thenfo(x) < 1. Hence, S is wholly contained in the interior of So. Nevertheless, as we shall prove in sec. 28.1, S has the same critical determinant as S o . Thus a critical lattice for So is also critical for S, whereas it has no point on the boundary of S.

It is not true that a critical lattice necessarily contains n independent points x with 1 5 f ( x ) < 1 + E . To prove this, Rogers [26a] constructs a star body Snin R" with the following properties: 1) Y is the only critical lattice, 2) each admissible lattice A with determinant d(A) < 41.6 is homothetic to Y.


CH. 4


His construction is inductive (with respect to n); in particular, S, is the union of the two domains I ( X ~ ) 5 ~ x 1x 2 - (x# 1 I1. For further details and the proof we refer to the paper cited. Here we mention a number of consequences. 1. Let S,,($) be the union of S,, and the block

lxll 6 a, lxil


(i = 2 ,.

. ., n),

where 0 < 9 < 1 and a is a fixed number with 1 < a < (1.6)1'(2n). The star body S,,(9) has a unique critical lattice, namely aY; the only pair of points of cry lying on, or near to, the boundary of S,,(9) is the pair +ae'. 2. For all 8 with 0 < 9 < 1, the body S,,($) has critical determinant a". But, in the metric 6 defined by (25.3), b(Sn,S,,(9)) -+ 0 for 6 -+ 0. Thus the relation (25.4) does not hold for unbounded star bodies. 3. Again, let 1 a < (1.6)"('") and let S be the star body with distance function f given by (2). Take a linear transformation A such that A = aY is critical for A S and has no point on the boundary of AS, and consider the star body S' = S,, u AS. It has a unique critical lattice, and this lattice has no point on the boundary of S ' ; by a slight niodification of S', we can arrange that it has just rn pairs of points on the boundary (m > 0 arbitrary; compare Hejtmanek [26a]).


26.4. We now consider unbounded star bodies mapped onto itself by an infinite group of linear transformations. We give the following definitions (Mahler [25c], Davenport and Rogers [28a]).

Definition 2. A non-singular linear transformation x' = Sax is called an automorphism of S if SaS = S. The condition implies that d(S) = d(SaS) = ldet Sal d(S). Thus, if d(S) is finite, we necessarily have det Sa = k 1; if a lattice A is critical for S, then so is Q A . Consequently, d(S) is infinite if there exists an automorphism Sa of S with det Sa # k 1. An example is the star body Ixi(x,)Z1 5 1.

Definition 3. An unbounded star body S is called automorphic i f there exist a positive number p and an (infinite) group G of automorphisnzs o j S possessin'g the following property

B 26



1". for each point x E S, there exist an automorphism Sa E G, mch that IDXI P. It is called fully automorphic if, in addition, one has 2". for each point x # o and each number z > 0, there exists an automorphism Sa' E G, such that Isa'xl 2 z.


Remark. The definitions 2 and 3 make sense for more general point sets, e.g., asymmetric star bodies; but usually they are employed only in the case of (symmetric) star bodies. The simplest examples of fully automorphic star bodies are given by IF(x)[ 5 1, where F ( x ) is one of the following forms (3) F(x) = X l X Z




(4) F ( x ) = X l X Z . . . x,{ ( x r +1 )' ( 5 ) F ( x ) = (XI)'+



+ (x,+'Y} . .. { ( x ,



+ (x.)'}

( r 2 0,s




2 0, r+2s

= n)

(0 < r < n).


Automorphisms o f these bodies are given successively by the diagonal matrices Sa = (wiaij), with wlwz * * u), = k 1; the matrices Ja having elements wl, ...,to, and blocks or,01, ...,a,+,0,along the main diagonal (the other elements being zero), where o, * or(or+ * w , + , ) ~= 1 and 0, , . . ., 0, are 2 x 2-orthogonal matrices; the matrices of the form


,- -

where u), w2 = k 1, 0, is an r x r-orthogonal matrix and O2 is an (n- r ) x x (n-r)-orthogonal matrix. In the last case the critical determinant is infinite if n is sufficiently large (see sec. 42.4). An example of a star body in R3 which is automorphic is given by (6)


5 1,


5 1.

We come back to the main problem of this section. Using theorems 25.5 and 25.7 Mahler [25c] proves

Theorem 4. Let S be an unbounded automorphic star body of thefinite type. Then there exists an S-critical lattice A which has apoint on the boundary of s.


cn. 4


Proof. Let f be the distance function determined by S and let A be an S-critical lattice. Suppose that no point of A lies on the boundary of S. Then, by theorem 25.5, there exists a sequence of points x' E A, such that f(x') + 1 for r -+ 00. According to definition 3, there exist a positive number p and automorphisms a,, Q 2 , , . ., such that



( r = 1 , 2 , . . .A

We put A, = Q,A and y' = a,x' (r = 1, 2, . . .). Then A, is S-critical. Hence, the sequence {A,] is bounded, in the sense of definition 17.2. Moreover,

IY'I 6 B,


1 ( r + a), f(Ar) = 1. Now, by Mahler's selection theorem, there exists a subsequence {Ar,} converging to a lattice A,. Since the sequence {y'} is bounded, we may suppose that the subsequence { y"} also converges. Applying theorem 25.7 to the sequence {A,,] we find that f ( A , ) = 1 and that A , contains a point y withf(y) = 1. This proves the theorem. Further, Mahler gives refinements of theorem 25.5. He proves the following interesting and important +

Theorem 5. Let S be a fully automorphic star body of theJinite type, with distance function5 Then,for any E > 0, each S-critical lattice A contains infinitely many points x with 1 5 f (x) < 1 + E . Proof. We shall prove the more precise statement that for each positive integer k there exists a positive number zk with the following property


each lattice A of determinant d(A) = A ( S ) has at least k pairs of points + x z o in (1 + E ) s ( ' ~ ) .

In virtue of theorem 3, the statement is true for k = 1. Now suppose it holds for some value of k, with some rk. Suppose also that for each integer r > r, there exists a lattice A, of determinant d(A,) = A ( S ) which has at most k pairs of points kx # o in (1 E)S(').Then there are exactly k such pairs; they are contained in (1 +E)S"~'. The lattices A, are all admissible for a h e d neighbourhood of o (depending on k). Hence, by the selection theorem, a suitable subsequence {A,,} converges to a lattice A. This lattice A has determinant d(A) = A ( S ) and has no point in the domain* (l+~)(S\s"~'). Furthermore, it has exactly k



Confer the corollary to theorem 25.6.

% 26



pairs of points + x # o in (1 + E ) S ( ~ ~Let ) . + X Ibe one of these pairs. By hypothesis, S is fully automorphic. So there exists an automorphism of S, such that IQx'I > z k . The lattice BA has at most k-1 pairs of points & x # o in (1 e)Strk).This contradicts our assumption and therefore proves that there exists a number rkfl > 0, such that (7) holds with k, T replaced by k + 1, rkf1.Hence (7) holds for all k and appropriate constants zk. This proves the theorem.


We remark that in sec. 28.3 we shall derive sharper results for a more special class of star bodies. Strengthening the condition 2" in definition 3 Mahler finds that, for a certain special class of fully automorphic star bodies, each critical lattice contains at least p independent points x with 1 5 f ( x ) < 1+a; here E > 0 is arbitrary and p is a fixed integer with 2 6 p 6 n. The result holds, with p = n, in the case of the star body So (sec. 25.1). For automorphic, and even for fully automorphic star bodies, it is not generally true that each critical lattice has a point on the boundary. Cassels [26a] proves this by taking the two-dimensional star body S given by Ixlx,l

6 l+&(sinm-l),




= 410g -

It admits the automorphisms a":x i = 2"x1, x; = 2-"x2 (n integral), is contained in So: Ixlx21 5 1 and satisfies d(S) = A(&). The lattice generated by the points (- 1, 1) and (9,$-I), where 9 = &(I +,/5), is critical for S, but has no point on the boundary of S.

26.5. Swinnerton-Dyer [26b] derives an analogue of theorem 3 for the inhomogeneous determinants (sec. 23) of certain domains which are automorphic in the sense of definition 2 and are very similar to the domain given by 5 x l x z * - x, Ia2 (a1,cc2 real numbers with a1 < u2). Here, for simplicity, we restrict ourselves to star bodies. If S is a star body and f the corresponding distance function, we denote by R(E) the set of points x with 1 5 f (x) 6 1S E ( E 2 0). In particular, R(0) is the boundary of S. We prove


Theorem 6. Let S be an unbounded automorphic star body. Suppose that, for each /? > 0, there exist positive numbers E , 6, such that each straight





line passing through a point z E R ( E ) , with Iz1 5 p, has a segment of length 1 6 in common with S. Then d'(S) is not zero and, if A ' ( S ) isfinite, there exists a critical grid r which has a point in R(0). Proof. We may, clearly, suppose that d'(S) is finite. Then there are S-admissible grids, and d'(S) is the greatest lower bound of the determinants of these grids. Now a grid r is S-admissible if and only if p(S, r )2 1. It follows that there exists a sequence of grids r, such that p ( S , r,) = 1 for all r,

d ( f , ) + d'(S) for r



For r = 1,2, . . ., choose a point zr E r,, such that f(z') --f 1 for r co. Since S is automorphic and the sequence of numbers f ( z ' ) is bounded, there exist a positive number p and a sequence of automorphisms Sa, of S, such that lQrz'l 6 p for r = 1 , 2 , . . .. Put --f

r: = a,r,,

y' = Qrzr

( r = 1 , 2 , . . .).

Then I'; is admissible for S and d ( r : )+ d'(S) for r 4 co. Moreover, Y ' E ~ : , ly'l S




= 1 , 2 , . .),

f ( y r )+ 1

for r



From the last relation and the hypothesis of the theorem it follows that, for sufficiently large r, each straight line through y' has a segment of length 2 6 in common with S. Here 6 is a fixed positive number. Thus, since ri is admissible for S, the corresponding homogeneous lattice A, = r i - y r is admissible for the sphere Ca:1x1 5 6 . This has two consequences. First of all, d ( r i ) = d(A,) 2 d(Ca) and therefore d'(S) 2 d(Ca). Thus there is an explicit positive lower bound for d'(S). Secondly, the sequence of lattices A , is bounded. Applying Mahler's selection theorem and the boundedness of the sequence { y'} we find that there is a sequence of integers r i such that A,, converges to a lattice A and that yr*converges to some pointy. As in the proof of theorem 25.7, it follows that the grid r = A + y is admissible for S. Moreover, d(T) = d'(S) andf(y) = 1. This proves the theorem. Swinnerton-Dyer remarks that, for reasons of compactness, the second condition of the theorem holds (with an unspecified 6 ) if each straight line passing through a boundary point contains an inner point of S.

§ 27



26.6. We mention some properties of the set of admissible lattices of a given automorphic star body, or rather the set of lattices A for which the functionf(A) has a positive value. A simple generalization of theorem 4 is given by

Theorem 4'. Let S be an automorphic star body and let A be a lattice with E A' such that

f( A ) > 0. Then there exist a lattice A' and a point x' f(x') = f ( A ' ) = f ( A ) and d(A') = d(A).

The proof is very similar to that of theorem 4. For given S, the set of numbers d(A), A a lattice with f ( A ) = 1, may have a complicated structure (in sec. 43, a particular case is treated in detail). At any rate, this set is closed; this is easily proved by applying the selection theorem (Mahler [26a]). Rogers [26b] calls a lattice A almost periodic with respect to a given group G of linear transformations of R" if to each neighbourhood N of one can find a compact set H t G with the following property:

(8) if l2 E G, then Ol2A E N for some 0 E H. Among other things, he shows that in theorem 4' the lattice A' can be chosen so as to be almost periodic with respect to the group G of automorphisms of s.In many cases, (8) holds with OQA = A ; then A is called periodic. 27. Reducible and irreducible star bodies 27.1. We recall that star bodies are always supposed to be closed.

Definition 1. A star body S is called reducible if there exists a star body T which is properly contained in S and has critical determinant A(T) = A ( S ) . IJ'such a star body T does not exist, then S is called irreducible.

If a star body T has the property mentioned, then we write T < S. In this section we shall deduce some criteria for the irreducibility of a star body and also deal with some examples. Below we shall prove the following result of Mahler [25d].

Theorem 1. If a star body S of ths$nite type is irreducible, then each boundmy point belongs to some S-critical lattice.


cn. 4


Mahler also gives a sufficient condition for the irreducibility of a star body S. His idea was elaborated by Rogers [27a] who introduced the concept of an irreducible point. A point x E S is called reducible or irreducible according to whether or not there is a star body T < S with x 4 T. If a point x E S is reducible, then so is each point of S in a certain neighbourhood of x . Hence, the set of irreduciblepoints is closed. We now have

Theorem 2. Let S be a star body of the jinite type. Then S is irreducible if and only if each boundary point of S is irreducible. Furthermore, a boundary point x of S is irreducible if and only i f ; for each neighbourhood N of the point x ; there exists a lattice A of determinant d ( A ) < A ( S ) which is admissible for S\(N v (-N)). Proof. If S is irreducible, then clearly each bmndary point of S is irreducible. Now suppose that S is reducible; let T be a star body with T < S. We show that T S\ contains a boundary point of S. Let x be an arbitrary point of S\T. Take a neighbourhood G(x. E ) = = ( y : Iy-xl < E } ( E > 0) which is disjoint with T and define a set V ( x , E ) by putting V(x, E) =


pG(x, E ) = { y : ly-pxl


< PE for some p 2 11.

The set V ( X , E is ) disjoint with T. It is an infinite convex cone which has the half-line of points px, p > 1 - E as a semi-axis and contains arbitrarily large spheres. The last property entails d ( V ( x , c)) = m, so that V ( x , E J is not entirely contained in S. On the other hand, it contains the point x E S. From these remarks it follows that V ( x , E ) contains a boundary point of S (consider the intersection of S and the segment joining x and some point y E V ( x , E)\S). This boundary point belongs to S T\ and is a reducible point of S. Next, let x be an irreducible boundary point of S and let N be a small neighbourhood of x (away from the origin). Letf be the distance function determined by S. By enlarging the values of the function f in a small double cone with vertex at o and containing x, we can get a continuous distance function g , such that the star body T determined by g has the following properties:


x$T, S \ T c N u ( - N ) .

5 27


31 1

Since x is an irreducible point of S, this star body T satisfies the relation d(T) < d(S). Any T-critical lattice A has the properties stated. Finally, let x be a reducible boundary point of S, and let T be a star body with T < S, x 4 T. Then there exists a neighbourhood N of x, such that N and T are disjoint. Then no lattice A has the properties required in the theorem. This completes the proof of the theorem.

Proof of theorem 1. Let x be a boundary point of S and let N, be the set of points y with Iy-xl < r-' ( r = 1,2, . . .). By theorem 2, the point x is irreducible; moreover, for each r, there exists a lattice A, such that d(A,) < d(S) and A, is admissible for S\(N, u ( - N , ) ) . Hence A, contains a point x' E N,. In virtue of the selection theorem, there exists a subsequence {A,,) converging to a lattice A, say. This lattice A is Sadmissible, by the corollary to theorem 25.6. Further, x = lim?' is a point of A, and d ( A ) g d(S). From this the theorem follows. 27.2. Mahler [25d] gives a satisfactory answer to the question which convex star bDdies in R2 are irreducible, in the sense of definition 1. His considerations are as follows. Let K be a bounded o-symmetric convex domain in R2.By lemma 22.2, a K-critical lattice A has 6 points on the boundary; if there are 8 points on the boundary, then, necessarily, K is a parallelogram and A is the lattice generated by the vertices and the mid-points of the sides of this parallelogram. One has the following

Theorem 3. If A is critical for K and has just 6 points on the boundary of K, then these point& are all irreducible.

Proof. The points of A on the boundary of K are given by three pairs +Y, with x 3 = xz-xl (sec. 22.1); moreover, (xl, x2} is a basis of A . The three points x', xz, x2+x3 are allinear, as well as the three points -xl, x 3 , x 2 + x 3 . Hence, since f x l , x2, x 3 lie on the boundary of K, the segments joining x z , x 2 + x 3 and x 3 , x 2 + x 3 , respectively, do not contain inner points of K. Now consider variable points y 2 = x2+9x3, y 3 = x3+9x2


< 9 < 1)

on these line-segments. We have y2 -y 3 = (1 - 9)(xz -x 3 ) = (1 -9)x'. The lattice A, generated by y 2 , y 3 has determinant d(As)= (1 - S2)d(A)< < d(A) and, for 0 < 9 < *, the points & ( y ' - y 3 ) are the only points


cn. 4


# o of A, in the interior of K. Hence, by theorem 2, the point x' is

irreducible. In a similar way one proves that the points x 2 , x 3 are irreducible.

As a corollary of theorems 1 and 3, we have Theorem 4. A bounded convex star body K in R2 is irreducible if and only if each boundary point is a point of a K-critical lattice. By this theorem, parallelograms and circular discs, with centre at are irreducible.


We recall (sec. 22.1) that a K-critical lattice determines an inscribed affinely regular hexagon of minimum area, and conversely. So we have

Theorem 5. A bounded convex star body K in R2 is irreducible $and only i f ull inscribed afinely regular hexagons have equal area. This theorem enables us to construct arbitrarily many irreducible domains in R2,in the following way. Take three points a', a,, a3, with a3 = a' -a' and det (a', a') > 0, and two arcs R , , R2 not containing straight line-segments and joining a', a' and a', a 3 , respectively. For x1 E R, ,let x2 be a point on R, satisfying det (xl, x2) = det (a', a')


det (a', a3),

such that x3 = x2 -xl describes an arc A3 joining a3, - a 1 . Then, if the domain bounded by the arcs fR , , fR , , fR3 is convex, this domain is irreducible.

27.3. Mahler also derives an analogue of theorem 3 for 3-dimensional convex star bodies. Later, we shall see that a critical lattice of such a body has at least 12 points on the boundary. Mahler proves

Theorem 6. Let K be a bounded convex star body in R 3 , ar2d let A be a critical lattice. Suppose that A hasjust 12points on the boundary. Then these points are all irreducible. With the help of this theorem, one can show that cubes and spheres in R3, with centre at 0, are irreducible.

B 27



Ollerenshaw [27a] gives a class of irreducible convex domains of a higher dimension by proving

Theorem 7. Let K be an irreducible bounded convex star body in R2, with distance function$ Then the cylinder C in R" ( n > 2 ) consisting of the points x with f ( x , , x2)

s 1,


g 1

(i = 3 , .

. ., r l )

is irreducible.

Also, for n g 5, a sphere about o in R" is irreducible. 27.4. It is not that easy to find non-convex irreducible star bodies. Below we shall treat an example given by Mahler [27a]. First, we prove a lemma on singular lattices (Mahler [29b]).

Lemma 1. Let S be a bounded 2-dimensional star body, with boundary R. Let A be a singular extremal lattice of S containing points +XI,rt: x 2 E R. Then R is convex*) at the points xl, x 2 . Moreover, the straight lines L , , L, which pass through xl, x2 and are parallel to the vectors x 2 , xl, respectively, are (inner) tac-lines to R (we say that xl, x 2 satisljl the tac-line condition). Proof. If R contains a point x which belongs to a sufficiently small neighbourhood of x1 and lies between the lines f L , , then the lattice A' generated by x, x2 is S-admissible and has determinant d(A') < d(S). This is a contradiction, and so there does not exist a point x having the properties mentioned. Thus L1 , and similarly L,, is an inner tac-line to R. Now we consider the star body S1 given by IXlX2l




5 J5.

The boundary R consists of an arc R , of the hyperbola xlx2 = - 1 in the fourth quadrant, an arc R, of the hyperbola xlx2 = 1 in the first quadrant, two segments R2 and R, of the line x , + x 2 = and the reflections of these arcs and segments with respect to 0. We consider the


By this we mean that there are tac-lines to R which, in a certain neighbourhood of x1 (2), pass through the body S.


cki. 4


lattice A having a basis (x', x'} of one of the following types:

1 ) x1 E R1; x', xZ-x1 E R3 2) x1 = ( - ~ , ~ ) ; x ~ , ~ ~ - ~ x ~ E R ~ u R ~ . These lattices are all admissible for S and have determinant J 5 . Each point of R belongs to at least one of the lattices specified. Furthermore, it is geometrically clear that there are no S-admissiblelattices of a smaller determinant. It follows that d(S) = and that the lattices considered are just the S-critical lattices. We prove that S is irreducible. Let A be a lattice of the type considered not containing an end-point of an arc. It has points f x l , +xz, f x 3 on R. It is easily seen that x2, x3 do not satisfy the tac-line condition. Hence, by the lemma, A is not critical for any star body T c S not containing the points f x'. Therefore, by theorem 2, the point x1 is irreducible. Hence, by theorem 2, S is irreducible.


Further examples of non-convex irreducible star bodies in RZ bounded by segments and circular or hyperbolic arcs are mentioned by Mahler [25c]. An interesting example was constructed by Ollerenshaw [27b]. It is a star body S , satisfying S, < Si,where 5'; is given by lXlX2l

5 1,


s, 1.

The star body S, is bounded by 8 congruent hyperbolic arcs and is obtained by taking the union of some continuous sets of parallelograms oxyz, where x, z vary on the boundary of Si and the sides xy, zy touch the boundary. The star body S, is uniquely determined by the requirement s, < Si. See also sec. 19.4. Mullineux [27a] considers bounded portions of the star body - 1 5 x,x,, S a ( a > 0) and he shows that, even for the class of bounded irreducible star bodies S, the coefficient of concavity w ( S ) (sec. 2.2) and the quotient V(S)/d(S) are not uniformly bounded. No unbounded irreducible star bodies are known (Mahler [25d]). 27.5. Mahler [27b] proved that each bounded o-symmetric convex body K in R2 contains an o-symmetric convex body Ki which satisfies the retation Kl K and, as a star body, is irreducible. He obtained such a body by considering the collection of all o-symmetric convex bodies K < K and by applying to this collection the selectiontheorem of Blaschke.


8 27



It should be observed that the condition that K1 is irreducible is much more stringent than the following one: Kl does not properly contain a convex star body K2 with K2 < Kl. The last condition can be expressed by saying that Kl is irreducible among the (0-symmetric) convex bodies. In this weaker sense, one can easily prove an n-dimensional analogue of the result discussed. As an analogue of theorem 2, Woods [27a] derived a criterion for irreducibility among the o-symmetric convex bodies in R";instead of all boundary points of a given body KO,this criterion involves only the extremal points of KO,i.e., those boundary points of KO which do not belong to.the interior of any straight line segment on the boundary of KO.Woods also gave an example of a bounded o-symmetric convex body in R 3 which is irreducible among the convex bodies K in R3, but not among the star bodies in R 3 . The example is the truncated cube lxil 6 1 (i = 1,2, 3), Ixl +x2+x31 I 1 investigated earlier by Whitworth (sec. 31.3). Mahler [25d] also raised the question whether or not each bounded star body contains an irreducible star body T with T < S. A very complete answer to this question has been given by Rogers [27b]. He gives at first a counterexample and then shows that the question has to be answered in the affirmative for a somewhat larger class of ray sets. Actually, Rogers calls an o-symmetric closed ray set S a star set; if, in addition, o is an inner point of S and S is the closure of the interior of S, then S is called by him a proper star set. He derives some analogues of theorems 1 and 2, and then proves that each bounded star set S cantaining o as an interior point contains a proper star set T with the following property:

d(T) = d(S) and d(T') < d(T) for each star set T' c T. An analogous result holds for proper star sets whose boundary is defined by an algebraic equation. The counterexample meant above is related to the proper star set S' in R2 consisting of the points whose polar coordinates r,cp satisfy r 6 1

or x < c p < x + x ,

r s 2


or x + x . $ c p $ 2 n ,

where x = arc cos 4. It is irreducible 'among the star sets'. But it is not a


CH. 4


star body. By a suitable modification of small parts of the boundary of S' one gets a star body S 3 S' of the type desired (with d(S) = d(S')).

28. Reduction of automorphic star bodies 28.1. We consider (unbounded) automorphic star bodies S. We shall deal with two entirely diffirent types of reduction of such bodies. First, we shall approximate S k y star bodies T which behave like Sonly at B large distance from 0 . Next, we shall consider bounded parts of S. Davenport and Rogers [28a] give the following

Definition 1. Let S be an unbounded star body, with a group G of automorphisms. Then a star body s c S is said to generate S if, for each z > 0, there exists an automorphism f2E G such that

(s'"= s n cJ.



To give an example, we take the star body So considered in sec. 25.1 and the star body f 0given by

f o ( x ) = IXlX,





s 1,




(i = 1, . . ., n-1).

The relation (1) holds for S = So and any automorphism Sa of the form xi' = W X i


(i = 1,. .., n-l),

x:, = 01 -n xn

(0> 2).

If generates S and S is of the finite type (so that, necessarily, = d(S). A det l2 = f l), then A(3) 2 A(S(*)),for each z, and so more general result is given by


Theorem 1. Let S be an (unbounded) automorphic stur body of thefinite type, with a group G of automorphisms. Let T be contained in S and suppose that, for each E > 0 and each T > 0, there exists an automorphism Sa E G such that (3)

9'' c (1+&)QT.

Then d ( T ) = d(S).

Proof. We have A (2") 5 d ( S ) and d (S(r))_I (1 +&)"A ( T )for all E, z > 0. Using definition 1 we get the following

§ 28



Corollary. I f T c S and if, for each E > 0, there exists a star body generating S, with c (1 + s)T, then A ( T ) = A ( S ) .



As an application, we prove a statement made in sec. 26.3. Take = S o , let T be determined by the function f given by (26.2) and let ( i = 1,. . ., n-1) and ]x,l > 1, E > 0 be arbitrary. If ]xi!< E " ( " - ' ) then f o (x) < ~1x1,and so f ( x ) < (1 +E)fO(x). Furthermore, since f is continuous and f ( x ) = 0 whenever f o ( x ) = 0, we have f ( x ) < 1+ E if Ix,J 5 1 and I x J , . . ., lxm-ll are small. So there exists a number 6 > 0 with the property that


if f o ( x ) 5 1 and lxil < 6

f ( x ) < 1+E

( i = 1,

. . ., n-1).

In other words, (1 +e)T contains the star body defined by f o ( x ) S 1, lxil < 6 (i = 1 , . . ., n- 1). The last star body generates S o . So we may

conclude that the star bodies determined by the distance functionsf, f o , respectively, have equal critical determinant. It is possible to indicate a general type of variations of the distance function of a star body which do not affect its critical determinant. Thus Mahler [25d] proves

Theorem 2. Let S be an (unbounded) automorphic star body of the finite type, with distance function f. Let g be a distance function, such that, for each 6 > 0, there exists an automorphism Q of S with the following property: g(l2x) 5 6 if 1x1 5 1. Furthermore, let cp(c, q ) be a distance function on R2 with rp(1,O) = 1, rp(1, q )


f o r all q.

Then the star body T with distance function h(x) = cp( f ( x ) , g(x)) has critical determinant A(T) = A ( S ) . Proof. Let E > 0 and z > 0 be arbitrary. Since cp is continuous, there exists a number 6 > 0, such that q(<, q ) < 1 + if ~ It1 5 1, lql 6 8 . Further, there exists an automorphism Q of S, such that g(s2x)

5 6z-1

if 1x1 g 1, and so g(sZx)

S if

1x1 6 7.

Now take any point x E S(r).Then, firstly, 1x1 5 z, and so g(&) Secondly, x E S, hence Qx E S, or alsof (Sax) 5 1. Consequently,

h ( ~ x= ) c p ( j ( ~ x ) ,g(fix)) < 1 + E

if x E [email protected]).

5 6.





In other words, S(r) is contained in (l+&)D-'T. The assertion of the theorem now follows from theorem 1. From the last theorem, we can deduce the following result of Davenport and Rogers [28b].

Theorem 3. Let p be a positiue integer c n. Let g l ( x l , . . ., xp), gz(x1,. . ., xP), h(x,+, , . . ., xn) be continuous distance functions of p , p , n - p variables, respectively. Then the star bodies S and T defined by {gl(xl,






* *,

~ p ) * ){ h~( x p + l y

x p ) } " . (gZ(x1,



* *

5 1,







5 1,

respectively, have equal critical determinant.

Proof. Put j ( x ) = (gl(Xl7

* *

g(x) = { S l ( X l Y * v(t;, ?) = ( 5 " K n - P )



~ p ) > ~* {h(xp+ ' ~ 1y




- {92(xp+





1 > * * *>


-p ) h ,

1(n - p ) h .

Then the conditions imposed upon g ( x ) and q(t, q ) in the enunciation of theorem 2 are satisfied, with automorphisms S2 of the form

XI = oxi

( i = 1 , . .. , p ) ,

. ., n),

xi = [xi ( i = p + l , .

where OY'[*-~= 1. This proves theorem 3 in the case that S is of the finite type. Now suppose that S is of the infinite type. Note that the automorphisms D just considered have determinant 1. It is easy to see that theorems 1 and 2 remain true for star bodies of the infinite type, if we restrict ourselves t o such automorphisms. Hence, theorem 3 is generally true. An application of theorem 3 is that the star bodies (4)

Ixlxz-*-xnI S 1 and ~{(xl)2+~~~+(xn)z)(x1)2(xz)z~~~(xn-1)zl 61

have equal critical determinant. The second star body is contained in the interior of the first one. 28.2. As to the second type ofreduction, Mahler [25d]gives the following

8 28



Dewtion 2. An unbounded star body S is called boundedly reducible, i f there exists a bounded star body T c S, such that A(T) = A(S). It is called boundedly irreducible if there does not exist such a star body T. Davenport and Rogers [28a] work with a stronger condition:

Definition 3. An unbounded star body S is called fully reducible, i f there exists a bounded star body T c S such that A(T) = A ( S ) and, moreover, T and S have the same set of critical lattices. The examples dealt with earlier in this section show that there exist boundedly irreducible star bodies. If Tis contained in the interior of S and A ( T ) = d(S), then each bounded portion T(') has critical determinant A(T('))< A ( S ) , and so T is boundedly irreducible. An example of a boundedly reducible star body may be obtained by taking the union of a bounded star body S and an unbounded star body S' with the property that some S-critical lattice is admissible for S'. Clearly, a boundedly reducible star body is of the finite type. As to the critical lattices of unbounded star bodies we have the following definitions:

Definition 4. A critical lattice A, of an unbounded star body S is called strongly critical, if there exist a boundedstar body T c S and a neighbourhood N of A , (in the space of lattices) such that (5)

d(A) 2 d(A,)

i f A E N and A is T-admissible.

Definition 5. A critical lattice A, of an unbounded star body S is called fully critical, if T c S (T bounded) and N can be chosen such that (6) either d(A) > d(A,) or A is S-critical, i f A E Nand A is T-admissible.

Mahler derives a sufficient condition for bounded reducibility. It is given by

Theorem 4. An unbounded star body S of the finite type is boundedly reducible, if each S-critical lattice is strongly critical.

Proof. Suppose that S is boundedly irreducible. Then we have A(S(')) < < d(S), for each T > 0. Let z run through the positive integers and let



CH. 4

A, be an S'')-critical lattice. In virtue of the selection theorem, there is a subsequence (A,,} converging to a lattice A,, say. Then d(A,) = A(S) and, by the corollary to theorem 25.6, A . is admissible for each S'", hence also for S. Hence, since d(A,,) < d(A,), ( 5 ) does not hold for any choice of T and N . Thus, A , is not strongly critical. This proves the theorem. In a similar way, Davenport and Rogers [28a] prove

Theorem 5. An unbounded star body S of the fnite type is fully reducible if each S-critical lattice isfully critical. An example of a fully reducible star body is given by - 1 5 xIx2 Ip

(p =- 0). There are also examples in R 3 and R4 (sec. 45.4).

28.3. Finally, we deal with reduction of automorphic star bodies to bounded star bodies. We denote the interior of S by int S. Davenport

and Rogers [28a] prove

Theorem 6. Let S be a fhlly automorphic star body of thefinite type, and let A be a lattice. Then the following assertions hold 1". is d(A) c A ( S ) , then int S contains infinitely many points of A; 2". if S is boundedly reducilbe and d ( A ) 5 A ( S ) , then S contains infinitely many points of A ; 3". if S is fully reducible, d ( A ) 5 A ( S ) and A is not S-critical, then int S contains infinitely many points of A. Proof. The assertion 1" is closely related to theorem 26.5. More precisely, if in assertion (26.7), S is replaced by (1 - q)S, and E , q are determined by A ( ( 1 - q ) S ) = d(A), I + E = (l-q)-', then that assertion takes the following form:

each lattice A of determinant d ( A ) < A ( S ) has at least k pairs ofpoints f x z o in Pk). Since (7) holds for each k , the assertion 1" follows.


The proofs of the assertions 2" and 3" are very similar; under the respective conditions, there exists for each positive integer k a positive integer zk such that S @ k ) or intS(Q), respectively, contains at least k pairs of points of A.

B 28




If S is generated by then each domain S(Tk)is contained in for a suitable automorphism Sak of S. This leads to the following refinement of theorem 6.

Theorem 7. Let S be a fully automorphic star body of thefinite type. Let S be generated by f and let A be a lattice. Then the assertions lo, 2" and 3" of theorem 6 remain valid $, in the colzclusions stated there, S is replaced by s. We saw already that there exist fully automorphic star bodies possessing a critical lattice no point of which lies on the boundary (sec. 26.4). Such a body necessarily is boundedly irreducible. Another example is obtained as follows. For 1 < B < 2, the star body S, given by - 1 5 x, x2 S fl satisfies


A(s,)2 min ( 4 4 +48, ~ 4 5 )

(sec. 44.1). Hence, if p is sufficiently near to 1 (more precisely, if 1 < p if ( 2 +,/24)), then d(S,) 2 J S . Consequently, for these values of By the star bodies given by - 1 x1x2 Ij3 and -8 5 x, x2 6 p, respectively, both have critical determinant BJ 5 . It is easily seen that the first one is boundedly irreducible. But it is also fully automorphic. Theorem 7 can be generalized for automorphic star bodies. Let S be an automorphic star body of the finite type. Let generate S and let 0 be an automorphism of S such that 0s c 5. Then the intersection


qs,q =kn 0's = 0


is called a core of S. For example, the infinite rectangle in R 3 given by S: (x1x21I1, lxglI 1; it is obtained by taking x2 = 0, lx31 I 1 is a core of the star body

S: Ixl x21 5 1, lx21 s I, lx31 5 1;

0:X; = 2x, ,X; = 4x2,X; = x3.

Davenport and Rogers [28a] now prove the following

Theorem 8. Let S be an automorphic star body of the finite type. Let C = C ( s ,0)be a core of S. Then the conclusions of theorem I are valid



CH. 4

i f A is subjected to the additional condition that it does not contain a point # o of C.

Corollary. If S is an automorphic star body of the finite type and i f A is a lattice with d(A) c A(S), then A contains either an infinity of points of int S, or at least one point # o of each core of S.



The general theory of star bodies which was strongly developed at the end of and during the years after World War I1 has not seen much progress in recent years. The high expectations that results on automorphic star bodies might help in the determination of lattice constants were fulfilled only to a very small extent. This probably brought the fall of interest in this area. A number of related questions received some attention in more recent time. These questions are dealt with in other supplementary sections, e.g. section viii (reducibility questions) and section ix (reducibility questions and semi-continuity results). The situation is different with the problem of lattice points on the boundary of compact star bodies or convex bodies. In section x we consider such questions. In particular results on balls and lattice octahedra are quoted. The results on balls in section x are related to sections xi and xii on packing of balls and on extreme (positive quadratic) forms. x. Boundary lattice points

In this section the number and configurations of points of an admissible or critical lattice on the boundary of a given compact convex body or compact star body will be discussed. Also general packings are considered. The material to be presented in this section is closely related to and supplements sect. viii, xi on packing and on packings of balls, and also sect. 26. For basic notions such as admissible and critical lattices, critical determinants, lattice packing, packing lattices, packing, density, etc. see sect. 17.1, 20 and viii. Let C be a compact subset of R", in particular a compact convex body or a (not necessarily symmetric) compact star body. A translate C y, y # o of C is called a neighbour of C if C and C +y have common boundary points but disjoint interiors. Let C be such that the interior of the difference set C-C of C (sect. l.l)ocoincideswith the difference set of the interior of C (which certainly is the case if C is a compact star body) and let A be a lattice.




CH. 4

Then the following two statements hold: (i) ( C + x : x E A ) is a lattice packing of C (of maximal density) if and only if A is admissible (resp. critical) for C - C. (ii) If {C x :x E A ) is a lattice packing of C, then the number of neighbours of C in this packing is equal to the number of points of A on the boundary of C - C. These statements have the effect that many of the properties discussed in this section admit two different interpretations. In the following we shall choose the interpretation which seems more appropriate.


x.1. Various estimates for the number of points of an admissible

lattice A on the boundary 2f an o-symmetric compact convex body K are known. Minkowski [GZ] proved that the number of points of A on the boundary of K is at most 3"- 1, see also Hlawka [20a], Hadwiger [xa], Groemer [xa] and sect. 26.2. The bound 3"- 1 is attained if and only if K is a parallelepiped and A is generated by the vectors along the edges of *K, see Groemer [xa, b]. If K is strictly convex or smooth then 3"- 1 can be replaced by 2"+l-2 according to Minkowski [GZ] and Groemer [xc]. The case of extremal bodies is treated in sect. 12.3. Assume now that d ( A ) < 34(K), that is A does not differ too much from a critical lattice of K . Then a result of Groemer [viiia] shows that 3" - 2"(d(A)/V(K)) is an upper bound for the number of points of A on the boundary of K. Note that 2"d(A)/V(K) 2 1 by Minkowski's fundamental theorem. A generalization of a different type is due to Woods [9a] (sect. 9.4): A, be the successive minima of K with respect to A . Then Let ,Ilr..., the total number of points of A contained on the boundary of any of the bodies A,K, . . ., 2°K is at most 3"- 1. Here 3"- 1 can be replaced by 2"' - 2 if K is strictly convex. These results are essentially refinements of Minkowski's theorem. In the following some extensions will be presented. See sect. 2.2 for the definition of the concavity coefficient of a star body. Groemer [26a] proved that the number of boundary points of a compact star body S with concavity coefficient o which belong to an S-admissible lattice is at most




2"~(n)w"+o(w") as o -, GO, where o ( . ) can be chosen uniformly for all S . Let { C + x : x ~ / l ) be a lattice packing of a compact set C. Then a general result of Hadwiger [ixa] implies that the number of neighbours of C in this packing is at most (1)

(V(C-C+C)-V(C))/V(P n U (C+x:x~/l)),

where P is a (fundamental) cell of A (sect. 3.4). If C is an o-symmetric convex body (1) reduces to 3"- 1. Groemer [xa] shows that the maximal number of translates of aK (s( > 0) with mutually disjoint interiors which meet the boundary of K but not the interior is at most

The case a = 1 generalizes Minkowski's result discussed before to the non-lattice case. Next we give a lower bound and a sharper upper bound for the case of critical lattices resp. densest lattice packings. An elegant result of Swinnerton-Dyer [261] (sect. 26.2) says that a K-critical lattice contains at least n(n + 1) boundary points of K . This theorem is related to an old result of Korkin and Zolotarev [39a, b, c] and Voronoi [39a] on the number of minimum points of a perfect positive quadratic form; see. sect. 39.2. Gruber [xa] proved that all compact convex bodies from an open dense subset of the space of all compact convex bodies have at most 2n2 neighbours in any of their lattice packings of maximum density. A weaker version of this concerns critical lattices of o-symmetric compact convex bodies. It is not known whether 2n2 may be replaced by a smaller number or even by n ( n + l ) for n 2 3. A partial generalization of Swinnerton-Dyer's theorem was given by Smith [xa] : Let C be a compact set in R" such that C - C contains a neighbourhood of 0.Then there exists a lattice packing of C in which C has at least n(n+ 1) neighbours. A different generalization is due to Bantegnie [viia]. More detailed results are known for the Euclidean unit ball K , which are often formulated in terms of quadratic forms. We shall phrase them



CH. 4

in a more geometric language. Both the lattice and the non-lattice case will be considered. Watson [xb] proved that for n = 2, ...,9, a K,-admissible lattice contains at most 6, 12, 24, 40, 72, 126, 240, 272 boundary points of K, respectively. The lattices for which the upper bounds are attained are unique up to rotations. For n = 2,. . ., 8 these lattices are precisely the critical lattices of K,. The uniqueness of the critical lattices of K, for n 5 8 was verified by Vettinkin [xia] and others; see sect. xi. For lattices in dimension n = 2, ..., 9 for which the upper bound is not attained the number of lattice points on the boundary of K, is considerably smaller than the upper bound, see Watson [xd]. The results for n = 2 , 3 have been known for a long time but it is difficult to give precise citations. Presumably they were known already to Gregory, Newton, Lagrange and Gauss. The case n = 24 will be discussed below in a more general context and also in sect. xi. For general n, Watson [xb] gave an upper bound for the number of points of a K,-admissible lattice on the boundary of K,. It is probably very weak, even for n = 10, as Watson himself states. Watson based his consideration on estimates for the determinant of n linearly independent points of a K,-admissible lattice; see subsection 3 below. The maximal kissing or maximal contact or Newton number of K , in R" is the maximum number of nonoverlapping Euclidean balls of unit radius that can touch K,. (Note that the arrangements of balls are not necessarily taken from a lattice packing.) Excellent surveys of kissing numbers are the articles of Coxeter [xa] (containing an interesting historical account) and Sloane [xib], tables with upper and lower bounds for moderately large n are given in Leech and Sloane [xia], Odlyzko and Sloane [xia] and Sloane [xib]. The maximal kissing number of K , is known only for n = 2, 3, 8, 24. For n = 2 it is 6. For n = 3 the problem arose in a conversation in 1694 of Newton (who asserted that the correct value was 12) and Gregory (who bid for 13). The answer in favour of Newton was eventually found by Hoppe [xa] in 1874; see [xa], Coxeter [xa]. More recent proofs are due to Schiitte and Van der Waerden [xa] and Leech [xa]. Odlyzko and Sloane [xia], and independently Levenktein [xib], established that for n = 8,24 the maximal kissing numbers of K, were 240 and 196560, respectively. They used the so-called linear programming method (see LevenStein [xic] for a survey). An elegant proof for the case n = 8 is contained in Seidel [xib].




Bannai and Sloane [xa] showed that the maximizing configurations are unique up to rotations. They can be obtained easily from the critical lattice of K 8 in R 8 and from the Leech lattice in R24. Coxeter [xa] conjectured that asymptotically as n -, co n*ee-'2h-1)nt = 0.652049.. .. 2f(n-l)n3

is an upper bound for the maximal kissing number in R". This was proved by Boroczky [xviiia]. A smaller asymptotic upper bound is due to Kabatjanskfi and Levenstein [xia] : 20.401n+o(n)

as n

-, 00.

On the other hand Wyner [xia] found a lower bound of the form 2(1-fzlog3)n+o(n)

= 20.207581 ...n + o ( n )



for the maximal kissing number in R". The problem how to place k points on the boundary of K 3 such that their minimal mutual distance is maximal has attracted much interest; see Fejes Toth [RF] and Danzer [xa] for results and references. x.2. As in sect. iii a lattice polytope P is a compact convex polyhedron

all of whose vertices belong to the integer lattice Y. The number of points of Y in P resp. in the interior of P a r e denoted L ( P ) and f , ( P ) . Under the assumption that L ( P ) > 0, Scott [xa] proved that for n = 2 L ( P ) 5 3 L ( P )+ 7, and for general n, Hensley [xa] gave an explicitly computable upper estimate for L ( P ) in terms of e ( P ) . For fixed e ( P ) > 0 Hensley's upper bound grows somewhat faster than exp(expn) as n -,co. The proof of Hensley uses tools from Diophantine approximation. An example of Zaks, Perles and Wills [xa] shows that the upper bound of Hensley cannot be improved essentially. For o-symmetric P the results are different. In this case one can derive an upper bound for L(P) which is substantially smaller than Hensley's bound. In fact, a theorem of Blichfeldt (see sect. 9.4) (reproved by Hlawka [24a] and generalized by Betke [xa]) and a result




of Van der Corput (Theorem 7.1) lead to the estimate L ( P )= L ( P ) - 1 + 1 5 n ! l / ( P ) + n 5 2 " - ' n !

(L(P)+l ) + n

assuming that the lattice points in the interior of P are not coplanar. Konyagin and Sevast'yanov [xa] show that there is a constant E, depending only on n such that for each lattice polytope P the number of vertices does not exceed E,V(P)("-')jtn+ I).

x.3. The following subsection is devoted to the investigation of the configuration of points on the boundary of an o-symmetric compact convex body K belonging to a K-admissible lattice. The results achieved are by no means satisfactory. We shall consider two classes of bodies K , octahedra and balls. Let A be a lattice. Consider linearly independent points xl,. . ., x" f A. If A is admissible for the convex hull of { k x ' , . . ., k x " ) , the latter is called a (generalized) lattice octahedron of A . A lattice octahedron is free if o and the vertices are the only points of A contained in it. The index of a lattice octahedron of A is the index (see sect. 3.3) of the sublattice of A generated by the vertices of the lattice octahedron. Compare sect. 31, 38.4, 40.3. Denote by I, the maximum index of a lattice octahedron in A and by J , the maximum index of a free lattice octahedron. Clearly I , 2 J,. An estimate mentioned in sect. 40.3 yields I , < (a,n)"+$,where a,

+ __


= 0.279403...

as n



and a result stated below shows that 1 = 0.058549.. . as n 2xe For small values of n we have the following list J , > (P,n)f("+'),where



B,, +


2, J , = 1, I 3 = 4,J3 = 2, I ,


18, J4





due to Minkowski [DA], Brunngraber [xa], Wolff [31a], Mordell [31a] and Bantegnie [xa]. The case of lattice octahedra, such that A / M is cyclic, where M is the sublattice generated by the vertices, was investigated for n 55



$ X

by Laub [xa] and Bantegnie [xa, b] ; an arithmetic interpretation was given by Mordell [3la]. Next assume that A is admissible for K , and contains n linearly independent boundary points of K , . Let L,, denote the maximum index of lattice octahedra defined by such points. Obviously we have J , 2 L,. In sect. 38.4 the following bounds were given :

L, 2 ~ ( ++n)/ntfl, 1

for all sufficiently large


Here y, is Hermite's constant, Note that

f ( l +in)/nj" =

(P,,)[email protected]+') ,



-+ 1/(2ne).

Davenport and Watson [38a] showed that L,

= L3 =

1, L4 = L , = 2,



4, L7




= 16.

Their proof was simplified by Watson [xc]. The special case that A corresponds to perfect quadratic forms was investigated by Hofreiter [xa], Rankin [xa] and Watson [xa, c]. Define an equivalence relation for lattices which are K,-admissible and which contain n linearly independent boundary points of K , in the following way: Two such lattices, say A and M are of the same type if there are n linearly independent boundary points of K , in A, say x', ...,x", and likewise in M , say y', ..., y", such that a point of the form vlxl +... +v,x" is in A if and only if vly'+...vnyn is in M . The types have been determined up to n = 8 by Hofreiter [xa] (n 5 6), RySkov [xa] (n = 7) and Zaharova [xa] (n = 8). For n = 2,. . ., 8 there exist 1, 1, 2, 3, 6, 13 and 39 types, respectively. RySkov [xa] investigated more generally systems of n minimum vectors of a lattice A with respect to K,. He found that the possible 'types' are not essentially different from what occurs in the case when all minimum vectors have length 1. RySkov related the types to n-dimensional compact facets of A ( a ) ; see sect. v. x.4. For results on boundary lattice points related to higher minima

(sect. 9.5) and higher critical determinants (sect. 17.4) we refer to the articles of Bantegnie [viia] and Gruber [xa]. The minimum number of neigbours of a convex body in a so-called connected packing has been determined by Groemer [viiig], see sect. viii.5.