Exercise 27, § 1

A real normed space E is said to be strictly convex if all the points of the unit sphere are points of strict convexity (II, p. 87, exerc. 6) of the unit ball. Show that, for E to be strictly convex, it is necessary and sufficient that for all $x \neq 0$ in E and for all $y \in E$, there exists a unique scalar $\alpha$ such that $x$ is quasi-normal to $x + y$. (Observe that the mapping $t \mapsto \|tx + y\|$ is convex in $\mathbf{R}$.)