Exercise 16, § 7

a) Let A be an n-dimensional compact convex set in the space $\mathbf{R}^n$, normed with the Euclidean norm; for every $z \in S_{n-1}$, denote by $\rho(z)$ the upper bound of lengths of segments parallel to the vector z and contained in A. Show that there exist two points $u, v$ of A such that the segment with end points $u, v$ is parallel to z and is of length $\rho(z)$; deduce that there exist two support hyperplanes of A that are parallel and pass respectively through $u$ and $v$ (consider the set $A' = A + \rho(z) \, z$, and separate the sets A and $A'$ by a hyperplane). b) Let d be the lower bound of the distances between two parallel support hyperplanes of A; show that there exist two points $a, b$ of A such that $\|a - b\| = d$, and that the hyperplanes passing respectively through $a$ and $b$ and orthogonal to $a - b$, are support hyperplanes of A (use a)).