Exercise 32, § 1

Let E be a normed space (real or complex) of dimension $\geq 2$, such that, for every $\varepsilon$ such that $0 < \varepsilon < 2$, the relations $\|x\| = 1,\ \|y\| = 1,\ \|x - y\| \geq \varepsilon$ in E imply the inequality $\| \frac{1}{2}(x + y) \| \leq \left( 1 - \frac{\varepsilon^2}{4} \right)^{1/2}$. Show that the norm on E is prehilbertian. (Reduce to the case when E is real and 2-dimensional, and argue as in V, p. 61; exerc. 4, a).)