Exercise 2, Appendix 0

Let E be a Banach space, K a precompact subset of E which is not a single point, and d the diameter of K. Show that there exists a point $x_0 \in K$ and a number r such that $0 < r < d$, such that $\|x - x_0\| \leq r$ for all $x \in K$ (choose $\varepsilon > 0$ small enough, and n points $y_1, ..., y_n$ of K such that every point of K is at a distance < $\varepsilon$ from one of the $y_j$, and put $x_0 = \frac{1}{n}(y_1 + \cdots + y_n)$.) Deduce a new proof of the Ryll-Nardzewski theorem for convex, strongly compact sets in E.