Exercise 24, § 7

In $\mathbf{R}^n$, every intersection of finitely many closed half spaces (resp. of closed half-spaces determined by hyperplanes passing through the same point) is called a polyhedron (resp. a polyhedral cone). A convex set $C \subset \mathbf{R}^n$ is locally polyhedral in a point $x \in C$ if there is a neighbourhood $V$ of $x$ in $C$ which is a polyhedron. *a) Show that, if a closed convex set $C \subset \mathbf{R}^n$ is locally polyhedral at a point $x \in C$, then the cone with vertex $x$ generated by $C$ is polyhedral. *b) Show that a compact convex set in $\mathbf{R}^n$ that is locally polyhedral at each of its points is a polyhedron (use a). *c) Let $P \subset \mathbf{R}^n$ be a closed convex set with an interior point. Show that the following conditions are equivalent : α) $P$ is a polyhedron. β) $P$ has only a finite number of facets (II, p. 87, exerc. 3). γ) $P$ is the convex envelope of a set which is the union of finitely many points and finitely many closed half lines. (To show that $\alpha$ implies $\beta$) take $P$ as the intersection of the smallest possible number of closed half spaces, and show that the hyperplanes defining these half-spaces are generated by facets of dimension $n - 1$ of $P$. To show that $\beta$ implies $\gamma$), argue by induction on $n$. Finally, to see that $\gamma$ implies $\alpha$), consider the polar $P^\circ$ of $P$. *d) Show that every convex polyhedron $P$ can be written in the form $Q + C_P$, where $Q$ is a compact polyhedron and $C_P$ the asymptotic cone of $P$. A non compact polyhedron cannot be parabolic. *e) Show that every facet of a convex polyhedron is an ultrafacet (II, p. 88, exerc. 7, d)) (argue by induction on $n$).