Exercise 8, § 7

Let $E$ be a vector space of finite dimension $n$ and $A$ be a compact convex set in $E$. a) Let $H$ be a hyperplane in $E$. Show that in an open half-space determined by $H$ and containing at least one point of $A$, there exists a point of strict convexity of $A$ (II, p. 87, exerc. 6). (Consider, in $H$, a closed euclidean ball $C$ of dimension $n - 1$ and of sufficiently large radius that contains $H \cap A$, then the euclidean balls $B$ of dimension $n$ and of larger radius containing $A$ and such that $B \cap H = C$.) b) Show that $A$ is the closed convex envelope of the set of points of strict convexity (use a)). c) Show that every extremal point of $A$ is a cluster point of the set of points of $A$ of strict convexity. (Using b) and the exerc. 9, a) of II, p. 66, note that an extremal point is the limit of a sequence of points of the form $\sum_{i=0}^n \lambda_{im} x_{im}$, where $\lambda_{im} \geq 0$, $\sum_{i=0}^n \lambda_{im} = 1$ and the $x_{im}$ are points of strict convexity of $A$; next use the compactness of $A$.)