Exercise 15, § 7

Let A be a compact set in the normed space E. a) Show that the distance apart of two parallel support hyperplanes of A is at most equal to the diameter $\delta$ of A. b) Show that there exist pairs of points $(a, b)$ of A such that $\|a - b\| = \delta$; for such a pair of points, there exist two parallel support hyperplanes of A passing respectively through $a$ and $b$ and whose distance apart is $\delta$ (consider the closed ball of centre $a$ and radius $\delta$).