This wikipedia article states that a normed space $(H,\rVert \cdot \lVert)$ where $$ \lVert x-y \rVert \lVert z \rVert + \lVert y-z \rVert \lVert x \rVert \geq \lVert x-z \rVert \lVert y \rVert $$ for all $x,y,z \in H$ has a unique inner product inducing the norm $\rVert \cdot \lVert.$ How do you prove this fact?
I guess that the fact that the inequality $$ \lVert x+y \rVert ^2 + \lVert x-y \rVert ^2 \le 2\lVert x \rVert ^2 + 2 \lVert y \rVert ^2 $$ for all $x,y \in H$ implies the parallelogram law might help, though I'm not sure.