Exercise 32, § 1

Let $E$ be a normed space (real or complex) of dimension $\geqslant 2$, such that, for every $\varepsilon$ such that $0 < \varepsilon < 2$, the relations $\|x\| = 1,\ \|y\| = 1,\ \|x - y\| \geqslant \varepsilon$ in E imply the inequality $\| \frac{1}{2}(x + y) \| \leqslant \left( 1 - \frac{\varepsilon^2}{4} \right)^{1/2}$. Prove that the norm on E is prehilbertian. (Reduce to the case where E is real and of dimension 2, and reason as in V, p. 60, Exercise 4, a).)