Exercise 3, § 7

Let A be a convex set in a vector space E, and let $x$ be a point of A. We call the set formed by $x$, and the $y \neq x$ in A such that the line passing through $x$ and $y$ contains an open segment, that is contained in A and contains $x$, the facet of $x$ in A. The internal points relative to the linear variety generated by A (II, p. 26) (resp. the extremal points) of A are the points whose facet in A is equal to A (resp. is a single point). a) Show that the facet $F_x$ of a point $x \in A$ is the largest convex set $B \subset A$ such that $x$ is an internal point of B (relative to the linear variety generated by B). b) For every point $y \in F_x$, the facet $F_y$ of $y$ in A is identical with the facet of $y$ in $F_x$. In order that $F_y = F_x$, it is necessary and sufficient that $y$ is an internal point of $F_x$ (relative to the linear variety generated by $F_x$). Deduce that, if $F_x$ is finite dimensional, and if $y$ is a non-internal point of $F_x$ (relative to the linear variety generated by $F_x$), then the dimension of $F_y$ is strictly less than that of $F_x$. c) A linear variety V in E which meets A and is such that for every $x \in A \cap V$, every open segment contained in A and containing $x$, is necessarily contained in V, is called a support variety of A. Show that, for all $x \in A$, the linear variety M generated by the facet $F_x$ of $x$ in A is the smallest support variety of A which contains $x$, and that $M \cap A = F_x$. For every support variety V of A, the intersection $V \cap A$ is the facet in A of each of its internal points (relative to the linear variety generated by $V \cap A$). d) Let A and B be two convex sets in E. For every point $x \in A \cap B$, the facet of $x$ in $A \cap B$ is the intersection of the facets of $x$ in A and in B. e) Let B be a closed convex set in a Hausdorff topological vector space E, and let B contain a closed linear variety M of finite codimension $n$; then every facet in B of a point of B contains a closed linear variety of codimension $n$ (II, p. 67, exerc. 14, d). If A is a convex set then the facet in $A \cap B$ of a point $x$ in $A \cap B$ is of finite dimension if, and only if, the facet of $x$ in A is of finite dimension : further if $p$ and $q$ are the dimension of the facet of $x$ in A and of the facet of $x$ in $A \cap B$, then $p \leq q + n$. In particular if $x \in A \cap B$ is an extremal point of $A \cap B$, then its facet in A is of dimension $\leq n$. f) Deduce from e) that if A is compact, and V is a closed linear variety in E, of finite codimension $n$, then every extremal point of $V \cap A$ is a linear combination of at most $n + 1$ extremal points of A.