A VOLUME INEQUALITY CONCERNING SECTIONS OF
CONVEX SETS
MATHIEU MEYER
ABSTRACT
We give a lower estimate of the volume of a compact convex subset of R" in terms of the volumes of its
sections by n pairwise orthogonal affine hyperplanes.
1. Introduction and notations
Let C be a compact subset of Un and (Hv ..., //„) be a family of n pairwise
orthogonal affine hyperplanes of Un; L. H. Loomis and H. Whitney ([4], see also [2,
pp. 161-163]) have given an upper estimate of the volume \C\ of C in terms of the
volumes of its orthogonal projections P'C onto the hyperplanes Ht, 1 ^ / ^ n. They
proved that
It is clearly impossible to obtain a lower estimate; however, when C is a convex body,
replacing the projections I*C by the sections C, = C 0 H{, we prove the following
inequality:
with equality if and only if C is a (generalized) octahedron, that is, C is a polyhedron
with vertices lying on the perpendiculars to the hyperplanes //„ 1 ^ / ^ n, drawn from
S = n {Ht, 1 ^ / ^ «}. It may be observed that another kind of lower estimate of \C\
in terms of sections by hyperplanes was given by H. Busemann ([1]), with equality if
and only if C is an ellipsoid.
For the sake of simplicity, it will be convenient to change somewhat these
preliminary notations. Let (ex, ...,en) be an orthonormal basis of IR", and for any
subset / o f {1, ..., «}, let
If the cardinality |/| of/equals m,\ ^m^n-\,
C{I) will be considered as a compact
subset of Un~m. For a compact subset A of Uk, let \A\ be its volume,
\A\ = fik(A),
where /ik is the Lebesgue measure on R* associated to the given orthonormal
basis.
Received 2 October 1986; revised 2 March 1987.
1980 Mathematics Subject Classification 52A20, 52A40.
Bull. London Math. Soc. 20 (1988) 151-155
152
MATHIEU MEYER
Finally, we define, for 1 ^ k ^ n
[
/
\i/<jjni/*
k\( n ic(/)i
With this notation, the inequality stated before can be written vn(C) ^ un_i(C); the
truth of this will be established by proving:
n
THEOREM 1. For any convex body C of U , the sequence vk(C), 1 ^k^n,
is
increasing.
To prove Theorem 1 we shall first verify that vn(C) ^ yB_i(C) for a special class
of C, and using Steiner symmetrization, deduce the general inequalities. Further, we
shall elucidate the equality case (Theorem 2) and give some remarks, applications and
examples.
2. Main results
n
For Ait iel, a subset of U , we shall denote by conv(/4<5 iel) the convex hull of
Av iel.
1. Let B be a convex body in Un; suppose that, for \ ^i^n,
symmetric about the hyperplanes {xt = 0}; if Bt = 2? n {*, = 0}, we have
PROPOSITION
(n\\B\)1/n ^ [(/j-l)!(|£1 |...|5 n |) 1/n ] 1/n ~ 1
B is
0)
There is equality in (1) if and only if B is an octahedron with vertices {±X{ei} 1 ^ i'^
n), where At > 0.
Proof. Let RJ denote the closed positive cone {(xv ..., xn)\xt ^ 0, 1 < i ^ «}.
Let K= B(\Ul and Kt = K(]{xt = 0} = BtnK. By the symmetries, we have \B\ =
2n \K\ and \Bt\ = 2n-1|#J, 1 ^ i < «. Thus, to get (1), we have only to verify that
(/i!|#|)1/n ^ [(«-l)!(|/g...|/g) 1 / n ] 1 / n ~ 1 -
(2)
For any point M in K, and for \ ^i^n, the cone conv (Af, K{), with vertex M and
base K(, is contained in K. Moreover for i^j, conv (M, Kt) 0 conv (M, Kj) is
contained in the hyperplane through M containing {xi = xf = 0}; it follows that
|conv(M, K{) n conv(M, K})\ = 0.
Thus, for any MeK, M = (xlt ..., xn), we have
\K\ > |conv (M, K,)\ +... + |conv (M, Kn)\ = (1 /«) (x x |/q +... + xJ/iTJ)
(3)
and K is contained in the simplex L:
Thus |/q ^ (|# 1 |...|/g)- 1 (wl/q)"^!)-1, which is a simplified form of (2).
If (1) is an equality, there is also equality in (2), (4) and (3); thus K= L. The body
B is then an octahedron with vertices { + Xte(, 1 < / < n}, where A, = n|^l/|A^| (it may
be observed that \K(\ # 0, 1 < / ^ «; otherwise B would have an empty interior). The
converse is immediate.
VOLUME INEQUALITY CONCERNING SECTIONS OF CONVEX SETS
153
Proof of theorem 1. (a) It suffices to prove that for any n ^ 2, and for any compact
convex subset C of Un, we have vn(C) ^ un_i(C).
Let us suppose that this inequality is true, and let 2 ^ k ^ n — 1; then for any
/ c {1, ...,«}, |7| = n — k, C(J) may be considered as a subset of Uk. We then get
vk(C(J)) = (k\\C(J)\rk > v
that is,
It follows that
k\( n \c(j)\)
^(^-l)!1^-1 n
n
|J'|-n-*+
But we have
n [ n ia^)i)=f n
|7'|-n-fc+l
, .
, (ri\
.
,
,. / n \
and since/;
=(«-&+1)
, we get
\k)
\k—\)
\I/(*2,)-II/*-I
n
||-n-*+l
Mm)
/
=»*-i(c).
-I
(b) By (a), we have only to prove that vn(C) ^ vn_x(C), n ^ 2. Steiner
symmetrization (see, for instance, [3]) about the hyperplane {xi = 0} preserves \C\ and
\C}\, j # /; and it can only increase |CJ, replacing it with |P*C|. After « Steiner
symmetrizations about {^ = 0}, 1 ^ / ^ n, C becomes a body B, satisfying the
hypothesis of Proposition 1; we thus get
un(C) = vn(B) > v^B) > v^iC).
Before examining the equality case in Theorem 1, we need the following lemma,
which follows from an easy computation of volumes.
LEMMA.
Let C be a compact convex subset of Un, such that
(i) \Cn\ = \PnC\,
(ii) if C is the Steiner symmetrized set of C about {xn = 0} then C = conv (Xen,
— Xen, Cn), for some I > 0.
Then there exists n^-0 and v ^ 0, such that // — v = 2X and C = conv(jien, ven, C n ).
2. IfC is a convex body in Un, then vn(C) = vn_x{C) if and only if C is
an octahedron with vertices fit et, v4 ev 1 ^ / ^ n, where //< ^ 0, v( < 0 and |//J -f |v<| ^ 0.
THEOREM
Proof If vn(C) = vn_x(C), then in the n Steiner symmetrizations of Theorem
1(6), we have |i*C| = \Ct\, 1 ^ / < n; moreover, by Proposition 1, the result B of
these symmetrizations is a symmetric octahedron of the described form. We now have
to apply the Lemma n times. The converse is obtained by calculation.
154
MATHIEU MEYER
3. Remarks, applications and examples
REMARKS, (a) If the n hyperplanes (Ht) are not assumed to be pairwise
orthogonal, Theorem 1 may be generalized as follows.
is
Let n<#< = {0} and e'^^^H^O});
then the sequence vk(C), \^k^n,
increasing, where
v'k(Q = \k\( n
and if |/| = k, p(I) is the Euclidean norm in U® of the exterior product /\ ( 6 / e,'.
(b) If C is a symmetric convex body in Un, it may be considered as the unit ball
of a ^-dimensional normed space E. Let /" be Un, endowed with the norm: \\x\\i =
|xj + . . . + |x n |; then Theorem 2 may be read as follows. We have vn(C) = v^C)
if
and only if E is isometric to /?, by a linear isometry sending (e{) on the canonical basis
of/?.
(c) With the hypothesis of Theorem 2, if vk{C) = vk_y(C), for some
k,2^k^n,
then vk(C) = v^Q for anyy ^ k; but if k # n, we may have vn(C) ^ v^C);
take for
example C = conv(0, e15 e2, e3, A/) in (R3, where M = (a, a, a) and a > 1/3.
(a) Let (C15 ..., CB) be n compact convex subsets of Un such that
C , c R J f i {x{ = 0}, 1 ^ i ^ «; then if |CJ is taken in Un~\ we have
APPLICATIONS,
(«!|conv(C 0 1 ^ i ^ n)\y/n > [ ( « - 1)1(1^1... ICJ) 1 '-] 1 '- 1 .
(b) For « = 3, let C = conv(0, X, Y; Z, e15 e2, e3), where A' = (0, ylt z j , r =
(x2, 0, z2), Z = (x3, y3, 0) with ^ 4 - z , , x2 + z2, x3+y3^l
and ^ , z15 xt, z2, x3, ^ 3 ^
0. From v3(C) ^ y2(C) we get the following inequality, which is not so easy to prove
directly:
EXAMPLES, (a) Theorem 1 does not apply when C is not convex: for n = 2, take
C{a) = ([0, 1] x [0, a]) U ([0, a] x [0, 1]), when a > 0, a-»0.
(6) In Theorem 1, one cannot replace sections with projections: for n = 2, let
C'(a) = {(*,y)eU2+, a ^ x+y ^ 1}, a-* 1.
(c) It is a natural question to ask whether Theorem 1 is still true when replacing
geometric means with arithmetic means in the definition of vk(C). This is clearly false:
for n = 2, take C = conv(0, (a, 0), (0, b)), with |a| # \b\. However, in thinking of the
case n = 2, it may be asked if this substitution is possible if C satisfies the hypothesis
of Proposition 1 and {—1, 1 } " C C C [—1, l] n . The answer is no, but we have no
example for n < 12.
For n = 12, let
Then
and
f
^2
C = \ (xj, max{|xj, 1 ^ / ^ 7}+ £ l*y
y n (C) = 2(7!)1/12, i v x ( C ) = 2(7!56!7)1/132
VOLUME INEQUALITY CONCERNING SECTIONS OF CONVEX SETS
155
(d) Does the sequence vk(C) satisfy the inequality
vk(C)2k < i W C r V i C C ) * " 1 , Kk^n-\,
»0(C) = 1,
which would imply Theorem 1 ? The answer is no.
Let (blt ..., bu)eU* and C = {(x{)e[0, l] n , £{:?&«*< ^ *}• lt
denoting a+ = max (a, 0), that
L/c{l
is n o t
difficult to show,
n)
n
If 1 / 2 ^ 6 ^ 1 and Q = {(^)G[0, l] , x1 + ...+x n ^ I/ft}, we get i;fc
(l-kil-byy"1,
2^k^n,
and yx(Q) = 1. This gives the counterexample when
b-+\.
References
1.
2.
3.
4.
H. BUSEMANN, 'Volume in terms of concurrent cross sections', Pacific J. Math. 3 (1953) 1-12.
H. HADWIGER, Vorlesungen iiber Inhalt, Oberfldche und isoperimetrie (Springer, Berlin, 1957).
K. LEICHTSWEISS, Konvexe Mengen (Springer, Berlin, 1980).
L. H. LOOMIS and H. WHITNEY, 'An inequality related to the isoperimetrie inequality', Bull. Amer.
Math. Soc. 55 (1949) 961-2.
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