Exercise 7, § 7

Let $E$ be a vector space of finite dimension $n$ and let $A$ be a closed convex set in $E$ of which $0$ is an interior point. a) Let $F$ and $F'$ be two dual facets of $A$ and $A^\circ$ (exerc. 6); if $F$ is of dimension $p$ and $F'$ of dimension $q$, then show that $p + q \leq n - 1$. For every frontier point $x$ of $A$, the dimension of the facet of $x$ in $A$ is called the order of $x$, and the dimension of its dual facet in $A^\circ$ is called the class of $x$. The order (resp. the class) of a facet $F$ of $A$ is by definition the order (resp. the class) of one of the internal points of $F$ relative to the linear variety generated by $F$. An extremal point of $A$ is a point of order $0$; a smooth point of $A$ (exerc. 6) is a point of class $0$. b) A frontier point of $A$ of class $n - 1$ (and hence of order $0$) is called a vertex of $A$. Show that the set of vertices of $A$ is enumerable (consider the set of dual facets of the vertices of $A$, GT, VI, § 2, exerc. 12). c) Let $F$ be a $p$-dimensional facet of $A$, and $M$ a linear variety of dimension $n - p$, which meets $F$ in the single point $a$, such that $a$ is an internal point of $F$ and which contains an interior point of $A$. Show that, if $V$ is a support hyperplane of $M \cap A$ in $M$, that passes through $a$, then the hyperplane $H$ generated (in $E$) by $F \cup V$ is a support hyperplane of $A$. d) We say that a facet $F$ of $A$ of order $p$ and of class $q$ is an ultrafacet if $p + q = n - 1$; the dual facet is then also an ultrafacet of $A^\circ$. If a linear variety $M$ of dimension $n - p$ meets an ultrafacet $F$ in a single point that is an internal point of $F$ (relative to the linear variety generated by $F$), show that this point is a vertex of the convex set $M \cap A$, and conversely (use c)). Deduce that the set of ultrafacets of order $p$ of $A$ is enumerable. (Identify $E$ with $\mathbf{R}^n$, consider the projection of $A$ on each of the coordinate varieties of $\mathbf{R}^n$ of dimension $p$; if the set of ultrafacets of order $p$ of which the projection on $V$ is $p$-dimensional, is not enumerable, show that there exists a point of $V$ which is an interior point to a non-enumerable infinity of these projections considering the points of $V$ with rational coordinates; then use b).) Give an example of a convex set with a non-enumerable infinity of facets, each of which is not a single point nor an ultrafacet. e) If all the frontier points of $A$ are smooth, show that the mapping, which puts each point $x$ of the frontier $G$ of $A$ in correspondence with the unique point of the dual facet of $x$, is a continuous mapping of $G$ on the frontier of $A^\circ$ (cf. TG, I, § 9.1, corollary). In what case is this mapping bijective?