Exercise 27, § 1

One says that a real normed space $E$ is strictly convex if all the points of the unit sphere are points of strict convexity (II, p. 93, exerc. 6) of the unit ball. Prove that, for $E$ to be strictly convex, it is necessary and sufficient that for every $x \neq 0$ in $E$ and every $y \in E$, 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 $\mathbf{R}$.)