Exercise 35, § 1

We denote by E a finite-dimensional real Hilbert space of dimension $n$, by S the unit sphere of E and by $m$ the unique positive measure of norm 1 on S invariant under the automorphism group of E. We consider S as a metric space in which the distance is defined by $d(x, y) = \operatorname{Arc}\cos \langle x|y \rangle$. For $x \in S$ and every real number $r \geqslant 0$, we denote by $B(x, r)$ the set of points $y$ of $S$ such that $d(x, y) \leqslant r$; for every subset $A$ of $S$ and every real number $r \geqslant 0$, we denote by $A_r$ the set of points $x$ of $S$ such that $d(x, A) \leqslant r$.

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

b) Let $x_0$ be a point of $S$, $H$ the hyperplane of $E$ orthogonal to $x_0$, $x_1$ a point of $H$ and $\gamma$ the circular arc joining $x_0$ to $-x_0$ by passing through $x_1$, that is to say the set of points of $S$ of the form $x_0 \sin \theta + x_1 \cos \theta$ with $|\theta| \leqslant \pi/2$. For every $y \in \gamma$, we put $H_y = H + y$ and $S_y = S \cap H_y$; we denote by $m_y$ the unique positive measure of norm 1 on $S_y$ invariant under the automorphism group of $E$ which leaves $x_0$ fixed.

Let $A$ be a closed subset of $S$ and let $\gamma'$ be the set of points $y$ of $\gamma$ such that $A \cap S_y$ is nonempty. For every $y \in \gamma'$, there exists a unique real number $r(y)$ such that $0 \leqslant r(y) \leqslant \pi$ and that $m_y(A \cap S_y) = m_y(B(y, r(y)) \cap S_y)$; we denote by $s_{\gamma'}(A)$ the union of the sets $B(y, r(y)) \cap S_y$ for $y$ ranging over $\gamma'$. Now prove that $s_{\gamma'}(A)$ is closed and that one has $m(A) = m((s_{\gamma'}(A)))$.

c) For every closed subset $A$ of $S$, one calls the radius of $A$ the greatest lower bound $r(A)$ of the set of real numbers $r \geqslant 0$ for which there exists $x \in S$ with $A \subset B(x, r)$. We denote by $M(A)$ the set of closed subsets $C$ of $S$ such that $m(C) = m(A)$ and $m(C_\varepsilon) \leqslant m(A_\varepsilon)$ for every $\varepsilon > 0$. Now prove that the following conditions are equivalent for every pair $(A, B)$ of closed subsets of $S$:

(i) one has $m(A) = m(B)$ and $B$ is of the form $B(x, r)$ with $x \in S$ and $r \geqslant 0$; (ii) one has $B \in M(A)$ and $r(B) \leqslant r(C)$ for every subset $C$ of $A$ belonging to $M(A)$. (Reasoning by induction on $n$, one will 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 $A$.)