Centre for Discrete and Applicable Mathematics 

CDAM Research Report, LSECDAM9815June 1998 
Béla Bollobás and Graham R. Brightwell
Abstract
A convex corner is a compact convex downset of full dimension in R_{+}^{n}. Convex corners arise in graph theory, for instance as stable set polytopes of graphs. They are also natural objects of study in geometry, as they correspond to 1unconditional norms in an obvious way. In this paper, we study a parameter of convex corners, which we call the content, that is related to the volume. This parameter has appeared before, either implicitly or in special cases, both in geometry and in combinatorics, and a major aim of the paper is to expose connections between work in these two areas.
We define the content mu(S) of a convex corner S in R_{+}^{n} recursively by

As part of his proof of SaintRaymond's Inequality  Vol(S)Vol(S^{o}) >= 1/n!  Meyer effectively showed that mu(S) <= n! Vol(S) for any convex corner S. Sidorenko showed that, if S is the stable set polytope of the comparability graph of a partial order P, then mu(S) is the number of linear extensions of P and thus, via a result of Stanley, the content mu(S) is equal to n! Vol(S). We show that, if S is a 01 polytope, then mu(S) = n! Vol(S) if and only if S is the stable set polytope of a strongly perfect graph.
The other ingredient in Meyer's proof is that mu(S) mu(S^{o}) >= n! for any convex corner S and its polar S^{o}. We extend this result by working with the pointwise product of convex corners. The pointwise product of two sets A and B is A o B = { x o y : x in A, y in B }, where (x o y)_{i} = x_{i} y_{i}. We show that, if A, B and C are convex corners such that the pointwise product A o B is contained in C, then we have mu(A) <= mu(B^{o} ) mu(C). This leads to the following generalization of SaintRaymond's Inequality: if A_{1}, ..., A_{r} are any bounded sets in R_{+}^{n}, such that A_{1} o ... o A_{r} is contained in the l_{1} unit ball, then mu(A_{1}^{o}) ... mu(A_{r}^{o}) >= n! and so Vol(A_{1}^{o}) ... Vol(A_{r}^{o}) >= (n!)^{r+1}.
We prove various other results concerning content and volume of convex corners, and give counterexamples to two conjectures of Bollobás and Leader about pointwise products. We conclude with a number of open problems.
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