Characterization of normed spaces based on violation of parallelogram law

Question

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 $y \in X$, such that $x \neq \pm y$, the following holds: $$ \frac{1}{2}\frac{\|x+y\|^2 + \|x-y\|^2}{\|x\|^2 + \|y\|^2} < 1.$$

In other words, the parallelogram law holds only when the vectors are equal or opposite. 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.

Answer 1

If the above inequality holds for all nonzero $x,y$, then if all of $x,y,x+y,x-y$ are nonzero, we also have (applying your inequality to $x+y$ and $x-y$): $$ \frac{1}{2} \frac{\|2x\|^2 + \|2y\|^2}{\|x+y\|^2+\|x-y\|^2}\leq 1, $$ so $\|x+y\|^2+\|x-y\|^2 \leq 2(\|x\|^2+\|y\|^2) \leq \|x+y\|^2+\|x-y\|^2$, and we have the parallelogram identity. Of course, if $x-y=0$ or $x+y=0$, the parallelogram identity is trivial.