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.