Exercise 27, § 2

a) Let $B$ be a non-empty closed convex subset of $E$, a Hausdorff topological vector space, and let $X$ be a non-empty compact set in $E$. Show that if $A$ is a subset of $E$ such that $A + X \subset B + X$, then $A \subset B$ (if $a \in A$, consider a sequence $(x_n)$ of points of $X$ defined inductively by the relation $a + x_n = b_n + x_{n+1}$, where $b_n \in B$). Deduce that, if $A, B$ are two non-empty subsets of $E$, using the distance in $E$ and the procedure of GT, IX, § 2, exerc. 6. Show $A + X = B + X$ implies the relation $A = B$. b) Let $E$ be a normed space, $\sigma$ the distance function defined on the set $\mathfrak{F}(E)$ of closed non-empty subsets of $E$, using the distance in $E$ and the procedure of TG, IX, p. 91, exerc. 6. Show that if $A, B, C$ are three non-empty compact convex sets in $E$ then $\sigma(A + C, B + C) = \sigma(A, B)$ (if $S_\lambda$ is the ball defined by $\|x\| \leq \lambda$, note that $A + S_\lambda$ and $B + S_\lambda$ are closed convex sets and use $a$). c) Deduce from $a$ and $b$ that the set $\mathfrak{K}(E)$ of non-empty, compact, convex subsets of a normed space $E$, with the distance $\sigma$, can be identified with a cone in a normed space of which the laws of composition induce on $\mathfrak{K}(E)$ the laws $(A, B) \to A + B$ and $(\lambda, A) \to \lambda A$.