Exercise 35, § 1

Let E denote a real, finite dimensional hilbertian space of dimension $n$, S the unit sphere of E and m the unique positive measure of norm 1 on S which is invariant under the group of automorphisms of E. Consider S as a metric space in which the distance is defined by $d(x, y) = \operatorname{Arc} \cos \langle x|y \rangle$. For every $x \in S$ and $r \geq 0$, let $B(x, r)$ denote the set of all points $y$ in S such that $d(x, y) \leq r$; for every subset A of S and every real number $r \geq 0$, let $A_r$ be the set of all points $x$ in S such that $d(x, A) \leq r$.

a) Given two closed subsets A and B of S, let $\delta(A, B)$ be the lower bound of the set of all real numbers $r \geq 0$ such that $A \subset B_r$ and $B \subset A_r$. Show that $\delta$ is a distance on the set $\mathscr{F}$ of all closed subsets of S, and that $\mathscr{F}$ is a compact metric space for this distance. Show that the mapping $A \mapsto m(A)$ from $\mathscr{F}$ into $\mathbf{R}$ is upper semi-continuous.

b) Let $x_0$ be a point in S, let H be the hyperplane in E orthogonal to $x_0$, $x_1$ a point in H and $\gamma$ the arc of the circle joining $x_0$ to $-x_0$, passing through $x_1$, i.e. the set of all points in S of the form $x_0 \sin \theta + x_1 \cos \theta$ with $|\theta| \leq \pi/2$. For every $y \in \gamma$, let $H_y = H + y$ and $S_y = S \cap H_y$; let $m_y$ denote the unique positive measure of norm 1 on $S_y$ which is invariant under the group of all automorphisms of E which leave $x_0$ fixed.

Let A be a closed subset of S and $\gamma'$ the set of all points in $\gamma$ such that $A \cap S_{p'}$ is non-empty. For every $y \in \gamma'$, there exists a unique real number $r(y)$ such that $0 \leq r(y) \leq \pi$ and such that $m_y(A \cap S_y) = m_y(B(y, r(y)) \cap S_y)$; let $s_\gamma(A)$ be the union of the sets $B(y, r(y)) \cap S_y$ as $y$ ranges over $\gamma'$. Prove that $s_\gamma(A)$ is closed and that $m(A) = m(s_\gamma(A))$.

c) For every closed subset A of S, the infimum $r(A)$ of the set of all real numbers $r \geq 0$ for which there exists an $x \in S$ with $A \subset B(x, r)$ is called the radius of A. Let $M(A)$ denote the set of all closed subsets C of S such that $m(C) = m(A)$ and $m(C_\varepsilon) \leq m(A_\varepsilon)$ for every $\varepsilon > 0$. Show that the following conditions are equivalent for every pair (A, B) of closed subsets of S :

(i) $m(A) = m(B)$ and B is of the form $B(x, r)$ with $x \in S$ and $r \geq 0$; (ii) $B \in M(A)$ and $r(B) \leq r(C)$ for every subset C of A belonging to $M(A)$. (Arguing by induction on n, we deduce from b) that $s_\gamma(A)$ belongs to $M(A)$ for every closed subset A of S; if $r > 0$ is such that $A \subset B(x_1, r)$ show that every point of the boundary of $B(x_1, r)$ in S which belongs to $s_\gamma(A)$ also belongs to A.)