For a normed linear space $(X, \|\cdot\|)$, the Jordan-von Neumann theorem specifies when exactly the norm is induced via an inner product, namely when the parallelogram law is satisfied.

I would like to know if there is a reference for a normed linear space that violates the parallelogram law, but in only one direction. That is, for any non-zero $x \in X$ and $y \in X$, the following holds: $$ \frac{1}{2}\frac{\|x+y\|^2 + \|x-y\|^2}{\|x\|^2 + \|y\|^2} \leq 1.$$

Specifically, I'd like to know if such spaces, although they aren't Hilbert spaces, have nice properties that might be useful in their study.