Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-98-15

June 1998

Convex Bodies, Graphs and Partial Orders

Béla Bollobás and Graham R. Brightwell


A convex corner is a compact convex down-set 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 1-unconditional 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

mu(S)  =  
x in S  
xi mu(Si),
where  Si  is the projection of  S  on to the  ith  co-ordinate plane, and  mu({0}) = 1  for the zero-dimensional convex corner  {0}.

As part of his proof of Saint-Raymond's Inequality -  Vol(S)Vol(So) >= 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 0-1 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(Smu(So) >= n!  for any convex corner  S  and its polar  So.  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 = xi yi.  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(Bo ) mu(C).  This leads to the following generalization of Saint-Raymond's Inequality: if  A1, ..., Ar  are any bounded sets in  R+n,  such that  A1 o ... o Ar  is contained in the  l1  unit ball, then  mu(A1o) ... mu(Aro) >= n!  and so  Vol(A1o) ... Vol(Aro) >= (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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