Exercise 5, § 5

Let $A$ be a closed convex set in $\mathbf{R}^n$ and $x_0$ be a point of $\mathscr{C} A$; denote the euclidean distance in $\mathbf{R}^n$ by $d$. a) Show, without using th. 1 of II, p. 36, that there exists one and only one point $x \in A$ such that $d(x_0, x) = d(x_0, A)$, and that the hyperplane orthogonal to the line joining $x_0$ and $x$, and passing through $x$ is a support hyperplane of $A$. b) Deduce from a) a new proof of th. 1 of II, p. 36 when the space $E$ is finite dimensional. (Reduce to the case when $M$ is a frontier point $x_0$ of $A$; note that the lower bound of the distance of $x_0$ from support hyperplanes of $A$, is zero, and use the compactness of $S_{n-1}$.)