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.