Convex Bodies, Graphs and Partial Orders
Béla Bollobás, Graham R. Brightwel · Proceedings of the London Mathematical Society · 2000
A convex corner is a compact convex down-set of full dimension in Rn. 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 implicitly before: both in geometry, chiefly in a paper of Meyer (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 (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 pointwise product of two convex corners; our results enable us to give counter-examples to two conjectures of Bollobás and Leader Oper. Theory Adv. Appl. 77 (1995) 13–24) on pointwise products. 1991 Mathematics Subject Classification: 52C07, 51M25, 52B11, 05C60, 06A07.