Arch. Math., Vol. 55, 82-93 (1990) 0003-889X/90/5501-0082 $ 3.90/0 �9 1990 Birkh/iuser Verlag, Basel

On the Blaschke-Santal6 inequality

By

MATHIEU MEYER and ALAIN PAJOR

1. Introduction. Let K be a convex body in the n-dimensional Euclidean space E (= En). For z in int (K), the interior of K, let K z be the polar body of K with respect to z; denoting by ( , ) the scalar product, we have:

K z = {y + zly~E", (y,x - z) < 1 for every xeK}.

The volumic product p (K) of the body K is defined by

p(K) = inf {]K] ]KZ], z E int (K)}

where 1] denotes a volume measure on E, It should be noticed that p (K) is affine invariant, that is, for every affine isomorphism

T: E -~ E, we have p(TK) = p(K). As it is well-known (see Sections 2 and 3 below), this infimum is reached for a unique

point z = s (K), sometimes called the Santal6 point of K. It was proved by Blaschke ([2], 1923) for n = 2, 3 that p (K) < p (B), where B denotes

the Euclidean ball in E"; this result was extended by Santal6 ([16], 1948) for all values of n, with some restrictive hypothesis on the smoothness of the boundary of K (see [17]); however, it was shown later, by some rather technical arguments, that these assumptions can be dropped. The inequality

p(K) <= p(B)

is the so called Blaschke-Santal6 inequality. In 1981, Saint Raymond ([15]) gave a simple proof of this inequality, in the special case when K is centrally symmetric; namely he proved that in that case, p (K) is maximal if and only if K is an ellipsoid; some years later, Petty ([11], 1985) characterized in the same way the case of equality for general convex bodies.

Our aim in this work is to give a more general result than the Blaschke-Santal6 inequality, by a rather simple proof, using Steiner symmetrization; namely we prove the following:

Theorem. Let K be a convex body in E" and let H = {x ~ E"I (x, u) = a} be an affine hyperplane (u ~ E"\O, a ~ ~), such that int (K) c~ H + O. Then there exists z ~ int (K) c~ H such that

IK[ IK z] < p(B)/42(l - 2)

gol. 55, 1990 On the Blaschke-Santal6 inequality 83

where 2 ~ ]0, 1[ is defined by

I{xeKl (x ,u ) > a}l =,~IKI.

This paper is organized in the following way; in Section 2, we give a short proof of the Blaschke-Santal6 inequality for centrally symmetric bodies; in Section 3 we state and prove some technical lemmas and in Section 4, we prove the theorem quoted before and study the case of equality in the Blaschke-Santal6 inequality.

Finally, let us mention the problem of giving a lower bound to p (K); it was proved by Mahler ([8], 1909) for n = 2 that p (K) __> 8 and by Bourgain and Milman ([3], 1987) that there exist some c > 0 such that for every n and every convex body K of E",

p(K) > c"p(B).

However the problem of finding the exact value of

inf {p (K) IK convex body in E"}

is still open for n > 3. The infimum on special classes of convex bodies, of E" was obtained ([15], [12]) and the

extremal case characterized on these classes ([13], [14], [9], [5]).

2. The centrally symmetric case. When K is centrally symmetric, it is easy to verify (see Section 3 below) that the Santal6 point of K is the center of symmetry. Let H be an affine hyperplane and denote by P the orthogonal projection onto H.

Then the Steiner symmetral S (K, H) of K about H is defined by

S(K' H) = { xl - x2 + xl' x2 ~ K' y ~ H' Pxl = Px2 = Y}"

It is clear that S (K, H) is also a convex body such that

- S (K, H) is symmetric about H (that is 2 Px - x ~ S (K, H) whenever x ~ S (K, H)). - I S ( K , H ) I = IKI. - IfK is centrally symmetric (about z), then S(K, H) is also centrally symmetric (about

Pz).

For more details about Steiner symmetrization, see [7]. Our first lemma says that the volume product of a centrally symmetric convex body increases by Steiner symmetriza- tion about hyperplanes through its center of symmetry.

Lemma 1. Let K be a centrally symmetric body with center z and H be an affine hyperplane through z, then

[(S(K,H)) z] > [KZ[ and p(S(K,H)) > p(K).

P r o o f. By the preceding remarks, we have only to prove the first inequality. After an affine transformation, it may be supposed that z -- 0 and that H = {(xi)~"~[x . = 0} is the hyperplane of symmetrization. We identify E ( = P,.") with H x ]R and denote K 1 = S (K, H).

6*

84 M . M E Y E R a n d A . PAJOR ARCH. MATH

Let PK be the orthogonal projection of K onto H; then

K 1 = {(X, x) lX ~ PK, x = (x 1 - x2)/2, (X, xl) E K, i = 1, 2}

K ~ = {(Y, y) ~ H x R I (X, Y) + xy < 1, for X E PK and x such that

(x, x)~K}

K~ = {(Y, y) ~ H x ]R] (X, Y) + y" (xz - xz)/2 < 1, for X E PK and x i such

that (X, x,)~K, i = 1, 2}.

For a subset A of E ( = R " = H x R ) and y~lR, denote A(y)= {Y~H[(Y,y)~A}. Addition meaning here Minkowski sum, we have then

K~ + K~ c K~ for every yE~,. . 2

Observe that since K is centrally symmetric we have for every y ~ R , K~ = - K ~ it follows from the Brunn-Minkowski theorem applied in H that

(*) [ K ~ K~176 >lKO(y)l = 2 =

and by integration

IK~ = ~ [K~ dy > ~ IK~ dy = IK~ . []

P r o o f o f t h e S a n t a l 6 ' s i n e q u a l i t y f o r c e n t r a l l y s y m m e t r i c b o d i e s. As it is well-known, there exists a sequence (K,) of centrally symmetric convex bodies, converging in the sense of Hausdorff to 2B (where I2BI = ]KI) and such that K o = K and K, is a Steiner symmetral of K,_ 1, for n > 1. By the lemma, the sequence (p(K,)) is increasing and by continuity it converges to p(2B)= p(B). Thus

p(K) = p(Ko) < p(B). []

3. Preliminary results. Let K be a convex body in E.

We define a function f : int (K) ~ IR+, by

( )-" f(z)=[K z ] = v . ~ m a x ( y , x - z ) da(y)

Sn-1 ~N x~K

where v n = [B I denotes the volume of the Euclidean ball in E (= IR n) and a the rotation invariant probability measure on the sphere S"-1

It is clear that

(i) lira {f(z); z approaches the boundary of K} = + oo, (ii) f is strictly convex and differentiable on int (K).

Let F be an affine subspace of E such that int (K) n F 4 = 0, and define

F •

Vol. 55, 1990 On the Blaschke-Santal6 inequality 85

It follows from (i) and (ii) that f has a unique critical point z(K, F) on int ( K ) n F, and that this critical point is also a strict minimum o f f on int (K) n F. This allows to say that z = z (K, F) is characterized by the identities

z e int (K) n F, gradf(z) e F z

By the dominated convergence theorem, we have for z e int (K)

gradf(z)=nv, ~ f(max (y,x \-~.+1)

s.-1 \x~K -z>/! da(y).

The following lemma summarizes the preceding facts.

Lemma 2. Let K be a convex body in R" and F be an affine subspace of N" such that int (K) n F ~ O. Then there exists a unique point z = z(K, F) E int (K) n F such that one of the following equivalent properties holds.

(a) IK~l = rain {IKXl; x E int (K) n F};

(b) y 37 s"-l (max (y, x - z)) "+a da(y)eF •

(c) S y dy e v • K Z - 2

If F = E, then z (K, F) is the Santal6 point s (K) of K; in that case, Lemma 2 gives the well-known fact that s (K) is the unique point s in int (K) which is the center of mass of the body K s.

Dealing with Steiner symmetrization, we need some properties of stability which are given in three lemmas. The proof of the first two ones is very easy.

Before stating them, let us recall that if K is a convex body and z s int (K), then z e int (K z) and, by the bipolar theorem, we have (K~) z = K. We shall say that two affine hyperplanes are orthogonal if their respective orthogonal directions are orthogonal, and when speaking of symmetry about an affine hyperplane H, we shall mean always orthogonal symmetry.

Lemma 3. Let H 1 and H 2 be two orthogonal hyperplanes in ~"; if a body K is symmetric about H1, then the body S(K, H2) is symmetric about both H 1 and H 2.

Lemma 4. Suppose that a convex body K is symmetric about some hyperplane H through z e int (K) n H; then K z is also symmetric about H.

Lemma 5. Let K be a convex body, F an affine subspace of •" such that int (K) n F #: 0 and z = z (K, F). Let H be an affine hyperplane such that F c H and let L be the convex body defined by L z = S (K Z, H). Then we have

z(L, F) = z = z(K, F).

8 6 M . M E Y E R a n d A . PAJOR ARCH. MATH.

P r o of. It may be supposed that z = z ( K , F ) = 0 , that H = { x , = 0 } and that F = c~({x, = 0}; i = p + 1 . . . . . n) for some p, 1 < p < n - 1. By Lemma 2, we have

ofdyeF •

, ,i=, 1 IR " - l , K ~ iJi=l Denoting for x = [xdi=l ~ tx ~i=" ~ K~ it follows that

x i ]K~ for l <i<_p. R . - t

By the definition of Steiner symmetrization, we have ]K~ = ]L~ for every x e R " - ~; we get thus

xl]L~ for l <i<_p Rn- 1

which conversely gives

f d y E F l Lo

that is by Lemma 2, z (L, F) = 0 = z (K, F). []

For sake of completeness, we give a simple proof of the following lemma (see [10]), which is a particular case of a result due to K. Ball [1].

Lemma 6. [1]. Let f, 9, h : IR + -. IR + be three functions vanishing outside some closed intervals containing the origin, on which they are continuous and suppose that

(2xy'] Y > g(x) x+y h(y) x+y for every x,y > O. (*) f k,x + y/=

Then one has

+~ = 2 + + ~ " I f (t) dt g (t) dt ~ h (t) dt/ 0 0 0

+0o +oo

P r o o f . Let A = ~ f ( t ) dt, B = ~ O(t) dt, C= 0 0

+oo

I 0

h(t) dr; by continuity one

can define differentiable functions x, y: [0, 1] ~ R + such that

x (u) Bu= ~ g(t) dt,

0

y (u)

Cu= ~ h(t) dt. 0

We have B = x'(u)g(x(u)), C = y'(u)h(y(u)) for every u e [0, 1]. Thus if we set

2 x (u) y (u) t -

x (u) + y (u)'

Vol. 55, 1990 On the Blaschke-Santal6 inequality 87

we get:

A > Z ! f ~ t h ( y ) ~ j

But we have

(x + y)-2 du.

(**) er +(l--oO s>_r~s 1-~, 0 _ < c ~ < l , r,s>=O.

x Cx By By (*) and (**) applied to c~ = x+--y' r = ~ ) , s g ( ~ = we get

1

A > 2 ~ (Cx):'/x+'(By) "/x+y (x + y)-i dy. 0

Now by (**) applied to c~ - X

- - - , r = l/Cx, s = l/By we get x+y

A => 2 / ( ~ + 1 ) , which gives the result. []

4. The Blaschke-Santal6 inequality. Let K be a convex body and H an affine hyperplane separating the Euclidean space E into two half-spaces D+ and D_ ; let 0 < 2 < 1 ; we shall say that H is 2-separating for K if

ID+ n g l - l D _ n g l = 2(1 - 2) lgl 2

and when 2 = 1/2, we shall say that H is medial for K (we have then I D+ n g l = I D- n K[

= I g l / 2 ) .

It is easy to see that for every direction u ~ S"- 1 and every 2, 0 < 2 < 1, there exists at least one (and in fact two if2 4: 1/2) affine hyperplane H, orthogonal to u and Z-separating for K; it is then clear that int (K) n H 4: 0.

The following lemma generalizes Lemma I to general convex bodies.

Lemma 7. Let K be a convex body, H an affine hyperplane and z ~ int ( K ) n H; let 2, 0 < 2 < 1 such that H is 2-separating for K z. Then if K 1 is the Steiner symmetral of K with respect to H, we have

IK~I > 42(1 -- 2) lgzl .

P r o o f . We can suppose that H = { x , = 0 } and z = 0 ; let us use then the same notations as in Lemma 1.

Let x, y > O, X ~ K ~ (x) and Y ~ K ~ ( - y); then for every Z s R" - 1 and z ~ R such that (Z, z) ~ K we have:

(X,Z) + xz < 1

(Y, ZS-yz <= 1.

88 M, MEYER and A. PAJOR ARCH. MATH.

y X Now if T - X + - - Y, we get

x+y x+y

(T,Z) + <1 x+y =

for every Z ~ R " - 1 and z l, z 2 ~ R such that (Z, z~) ~ K, i = 1, 2. This means that

T e K ~ \ x + y /

Using Minkowski sum in N" - l, we get thus for every x, y > 0.

x KO (2xy~ Y KO(x) + (-y) = K o . x+y x + y \ x + y J

By the Brunn-Minkowski theorem in R " - i , denoting

f (z)=lK~ g(x)=lK~ h ( y ) = l K ~

we get formula (,) of Lemma 6. + Now, since by Lemma 3, K ~ is symmetric about H, we have 5 f(z) dz = ]K~ and

0

since H is 2-separating for K~ we have g (x) d h (y) d = 2 (1 - 2) [K~ It

is clear that the functions f, 9, h: [0, + oe[ --, IR+ satisfy the hypothesis of Lemma 6. Since +oo +oo

by definition of 9 and h, one has 5 9(x) dx + ~ h(y)dy = ]K~ we thus get 0 0

+ +f~ - 2 2(1 - 2 ) I / ~ IK~ = 2 9(x)dx h(y)dy 0 0

which gives the result. []

Theorem. Let K be a convex body in E, H be an affine hyperplane such that int (K) c~ H 4= 0 and suppose that H is 2-separating for K for some 2, 0 < 2 < 1. Then there exists z ~ int (K) c~ H such that

[KI IKZl <= v2/4 2(1 - 2) = p(B)/42(l -- 2).

P r o o f. We proceed by n successive Steiner symmetrizations until we get a centrally symmetric body.

Let u i e S"- i , ui orthogonal to H H i and let i=, S" - i = (ui)i= 2 c such that (u 1 . . . . , u.) form an orthonormal basis for E. Let z i = z (K, Hi), with the notations of Lemma 2, and define a body K i by the identity

K ? = S (K% Hi).

Vol. 55, 1990 On the Blaschke-Santal6 inequality 89

Then [K~ll = IK zl I. By Lemma 4, K 1 is symmetric about H 1 and by Lemma 7, applied to K ~1, z = z 1 and H = H1, 2-separating for K = (KZl) zl, we get

IKll > 42(1 - 2 ) Ig l and thus [KI[ I g~ l > 42(1 - 2) [g l IgZ~l.

Choose now the hyperplane H2, orthogonal to u2, and medial for K 1 and define

z2 = z(K1, H1 ~H2) .

Since by Lemma 5 we have z 1 = z(K, H1) = z(K 1, H1), we get

[K~21 = min {]K~[; z ~ H 1 c~ H2}

> min {IK~I; z EH1} = IK~a~'~l)l = IK~'I.

We define now a new convex body K 2 by the identity

Ig~21 = s(g~ ~, n2).

By Lemmas 3 and 4, K z is symmetric about both, H 1 and H 2. Since H 2 is medial for K 1 , we get by Lemma 7 applied to K~ ~, z = z 2 and H = H 2 that

Ig2l _-> IKll .

Moreover, we have

IK~=I = I s (g~ ~, n 2 ) [ - - I K ~ [ > [K~ll.

It follows that

Ig211K~I > IKll I g ~ l .

We continue this procedure by choosing hyperplanes H 2 . . . . . Hn, points z 2 . . . . . z,,, and defining convex bodies K 2 . . . . . K, such that for 2 < i < n, we have

(i) H i is medial for Ki-~ and orthogonal to ui;

(ii) z i = z ( K i _ l , H 1 nH2c~ ... ~H~); (iii) K F' z' = S(Ki-1 , Hi).

We prove then by induction that ([K,I IKF'I) is an increasing sequence, for 2 < i < n. From (ii) (iii) and Lemma 5, we have

z i = z ( K i _ l , H 1 n ... nHi) = z(Ki, H 1 n ... nHi) .

Choosing Hi+l, z,+l, K,+ 1 according to (i) (ii) (iii), we get thus

. . . . = (Ki ,H~+I)I Ig~'+*l [Ki+I [ IS . . . . =

= i n f { I K r l ; z e H 1 ~ ... n H i n H i + l }

> inf {]KT]; z ~ n I c~. . . n n , } = IK z(K''n . . . . . . n,~] = [KZ.[.

Now, Lemma 7 applied to K~', z = z i + 1 and Hi+ 1, medial for Ki+ 1 = (K; '+ 1) .... , gives:

Ig/+l[ _-> IKil.

90 M . MEYER a n d A . PAJOR ARCH. MATH.

Thus

42(1 -- 2)]K[ IKZ' I <= IKll IK~I _-< [K2I ]K~2[ < . . . < IKnl IKX"I

from Lemmas 3 and 4 we get that K i is symmetric about H~, 1 _< j __< i. Thus K, is symmetric about Hi, 1 _< i < n. It follows that the point z,(=H~ n H 2 n . . . n i l , ) is a center of symmetry for K,; one has now to apply the Blaschke-Santal6 inequality for centrally symmetric convex bodies (Section 2) to get the result.

Corollary (Blaschke, Santal6, Saint Raymond, Petty). Let K be a convex body in the Euclidean space E; then p(K) < p (B), with equality if and only if K is an ellipsoid.

P r o o f. The inequality p (K) < p (B) follows from the preceding theorem applied to some hyperplane H medial for K. By the affine invariance, we have p (g) = p (B) for any ellipsoid g of E.

Suppose now that p (K) = p (B) and let s be the Santal6 point of K; by the theorem, every hyperplane which is medial for K contains a point z(H) in H such that

IKI IKZ*ml _-< p(B)

We have thus

p(K) = IKI IKS[ _-< [KI IK~(ml < p(B)

and by the uniqueness of the minimum of f (z) on int (K) (see the beginning of Section 3), we get

z (H) = s, for every medial hyperplane H for K.

It follows that s belongs to every medial hyperplane for K. Thus, since for every direction u ~ S"-1 there exists a medial hyperplane for K orthogonal to u, every hyperplane through s is medial for K. It follows now from the Funk-Hecke theorem (as applied in [4]) that s is a center of symmetry for K. We have now to refer to the simple and elegant proof of the case of equality p (K) = p (B) for centrally symmetric bodies, given by Saint Raymond [15] to get that K is an ellipsoid (see the appendix).

R e m a r k. The volume product p (K) may be equivalent to p (B) and yet the body K be far from any ellipsoid as is shown by the following example communicated to us by

K. Ball. Take t such that tlogn > 1, define t. = and K. = [ - t., t.]" n B. An

( c ) p(B)forsomeuniversalcon- elementary computation shows that p (K.) >-_ I nt/2 - i "

stant c > 0. On the other hand it is clear that for any ellipsoid d ~ such that g c K. c 2g then 2 > 1/t..

Appendix. We show that for a centrally symmetric body K, if p (K) = p (B), then K is an ellipsoid.

Suppose that p (K) = p(B); then there is equality in Lemma 1, for any hyperplane H through the center of symmetry; using the notations of this lemma, it follows from the

Vol. 55, 1990 On the Blaschke-Santal6 inequality 91

equality case in the Brunn-Minkowski theorem that

(i) K~ is a translate o f K ~ = -K~ K~ + K~ y)

(ii) K~ = 2

for every y such that K~ 4: O. Property (i) says that K ~ (y) has a center of symmetry, say Z(y). Thus if p(K) = p(B), every cross-section of K ~ by an affine hyperplane is centrally

symmetric. If n __> 3, this gives, by a theorem of Brunn (see [6]) that K ~ and thus K is an ellipsoid.

However, instead of using this deep result of Brunn, and to solve the problem also in dimension 2, a more precise version of the equality case in Lemma 1 can be given. The following lemma is implicit in [15].

Lemma 8. Under the hypothesis of Lemma 1, the following are equivalent

(a) IS(K, H)~ = [K~ . (b) Every cross section of K ~ by an hyperplane parallel to H has a center of symmetry,

and these centers of symmetry are in line. (c) The centers of all the cross-sections of K by lines orthogonal to H lie in a hyperplane.

P r o o f. By the bipolar theorem (b) and (c) are equivalent; they clearly imply (a); on the other side, to verify that (a) implies (c) we first reduce to the 2-dimensional case.

Indeed, with the preceding notations, it is enough to prove that for every X e PK, we have for some e depending on X,

<X,Z(y)) = c~y, for every yeI = {z;K~ 4: 0}.

Fix thus X e PK, X # O, and define ~o, ~ : I --, ~ by

~o(y) = max {<X, r>; Ye K~

~(y) = max {<X, Y>; Ye K~

By (i) and (ii), we have for every y e I.

q)(y)- (p (-- y) (iii) = <Z (y), X >,

2

(iv) t) (y) = q~ (y) + (p ( - y) 2

Define now two centrally symmetric convex bodies A and A 1 in ~.2 by

A = {(s, x) ~ ~.2; (sX, x) E K},

A1 = {(s, x) e~2 ; (sX, x) e K1}.

92 M. MEYER and A. PAJOR ARCH. MATH.

It is clear that A 1 is the Steiner symmetral of A about the second axis, and by the bipolar theorem we have,

A ~ = {(t,y) e ~ , x I ; - q g ( - y ) < t < cp(y)},

A ~ = { ( t , y ) ~ N ~ x I ; - ~ ( - y ) < t < ~(y)}.

It follows from (iv) that A ~ is the Steiner symmetral of A ~ about the first axis, and thus that IA~ = [A~ . If the lemma is proved in dimension 2, then all the centers

(Y, (P (Y) --2tP ( - Y)) of the chords of A~ orthogonal to the first axis are in line; thus f o r - -

some e e R , we have by (iii),

cp(y) -- ~p (-- y) (X ,Z(y)> = = ~y for every yE1 .

2

This concludes the reduction to the 2 dimensional case. We refer now to a lemma due to Saint Raymond ([15], Lemma 11).

Now if p (K) = p (B), we get by the preceding lemma that (c) is true for any hyperplane H; as it is classical, using for instance the unicity of the ellipsoid of maximal volume contained in K, it implies that K is an ellipsoid.

References [1] K. BALL, Logarithmically concave functions and sections of convex sets in R". Studia Math.

88, 68-84 (1988). [2] W. BLASCnrd~, Vorlesungen fiber Differentialgeometrie II. Berlin-Heidelberg-New York 1923. [3] J. BOLrRGArN and V. D. MILMAN, New volume ratio properties for convex symmetric bodies in

~". Invent. Math. 88, 3t9-340 (1987). [4] K. J. FALCONER, Applications of a result on spherical integration to the theory of convex sets.

Amer. Math. Monthly 90, 690-693 (1983). [5] Y. GORDON, M. MEYER and S. REISNER, Zonoids with minimal volume product. Proc. Amer.

Math. Soc. 104, 273-276 (1988). [6] P. M. GRUBER und J. H()BINGER, Kennzeichnung von Ellipsoiden mit Anwendungen. Jahrbuch

Oberblicke Math. 9-29 (1976). [7] K. LEICHTWEISS, Konvexe Mengen. Berlin-Heidelberg-New York 1980. [8] K. MAHLER, Ein Ubertragungsprinzip ffir konvexe K6rper. (~asopis P6st. Math. Fys. 68,

93-102 (1909). [9] M. MEYER, Une caract6risation volumique de certains espaces norm6s. Israel J. Math. 55,

317-326 (1986). [10] V. D. MILMAN and A. PAJOR, Isotropic position and inertia ellipsoids and zonoids of the unit

ball of a normed n-dimensional space. GAFA (Israel Functional Analysis Seminar 87/88), LNM 1376, 64-104, Berlin-Heidelberg-New York 1989.

[11] C. M. PETTY, Affine isoperimetric problems. Ann. New York Acad. Sci. 440, 113-127 (1985). [12] S. REISNER, Random polytopes and the volume product of symmetric convex bodies. Math.

Scand. 57, 386-392 (1985). [13] S. R.EISNER, Zonoids with minimal volume product. Math. Z. 192, 339-346 (1986). [14] S. REISNER, Minimal volume product in Banach spaces with a 1-unconditional basis. J. London

Math. Soc. 36, 126-136 (1987). [15] J. SAINT RAYMOND, Sur le volume des corps convexes sym6triques. S6minaire d'Initiation/t

l'Analyse, 1980-1981, Universit6 PARIS VI, Paris 1981.

Vol. 55, 1990 On the Blaschke-Santal6 inequality 93

[16] L. A. SANTAL6, Un invariante afin para los cuerpos convexos del espacio de n dimensiones. Portugal. Math. 8, 155-161 (1949).

[17] R. SCHNEIDER, Random polytopes generated by anisotropic hyperplanes. Bull. London Math. Soc. 14, 549-553 (1982).

Anschriften der Autoren:

Mathieu Meyer Equipe d'Analyse U.A. N ~ 754 au C.N.R.S. Universit6 Paris VI Tour 46 - 4 ~"~ Etage 4, Place Jussieu F-75252 - Paris Cedex 05

Eingegangen am 1.12. 1988

Alain Pajor Universit6 Paris VII U.ER. de Math6matiques Tour 4 5 - 5 5 5 ~'~ Etage 2, Place Jussieu F-75251 - Paris Cedex 05