CONVEX BODIES, GRAPHS AND PARTIAL ORDERS

Abstract

A {\em convex corner} is a compact convex down-set of full dimension in ${\mathbb R}_+^n$. 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 {\em content}, that is related to the volume. This parameter has appeared implicitly before: both in geometry, chiefly in a paper of Meyer ({\em Israel J.\ Math.} 55 (1986) 317--327) effectively using content to give a proof of Saint-Raymond's Inequality on the volume product of a convex corner, and in combinatorics, especially in a paper of Sidorenko ({\em Order} 8 (1991) 331--340) relating content to the number of linear extensions of a partial order. One of our main aims is to expose connections between work in these two areas. We prove many new results, giving in particular various generalizations of Saint-Raymond's Inequality. Content also behaves well under the operation of {\em pointwise product} of two convex corners; our results enable us to give counter-examples to two conjectures of Bollob\'as and Leader ({\em Oper.\ Theory Adv.\ Appl.} 77 (1995) 13--24) on pointwise products. 1991 Mathematics Subject Classification: 52C07, 51M25, 52B11, 05C60, 06A07.