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Does the parallelogram law for vectors of equal length imply the full parallelogram law? That is, if for all norm one vectors $x$ and $y$ in a Banach space $X$ it holds that $\lVert x-y\rVert^2+\lVert x+y\rVert^2=4$, does it follow that $X$ is isometric to a Hilbert space?

I suspect the answer is "no", but I cannot come up with an example. Of course, it is enough to consider the question in two dimensions.

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I will give it a try, based on Day's idea. Let $X$ be a two-dimensional Banach space with the given property and denote by $B_X$ its unit ball. Consider the ellipsoid of maximal volume (denoted by $B_2$) contained in $B_X$ (the John's ellipsoid) and denote by $\|\cdot\|_2$ the induced Euclidean norm. The goal is to show that $B_X=B_2$.

From John's theorem concerning the ellipsoid of maximal volume (or Loewner's Lemma for two dimensions in Day's paper), it follows that $B_X$ and $B_2$ have at least four contact points. Unless $B_X=B_2$, the contact points cannot form a dense subset of $B_X$. Assuming $B_X\neq B_2$, we can find contact points $x$ and $y$ such that $\displaystyle\frac{x+y}{\|x+y\|}\in B_X$ is not a contact point. Hence $\|x+y\|<\|x+y\|_2$. Therefore:

$$ 4=\|x+y\|^2+\|x-y\|^2<\|x+y\|_2^2+\|x-y\|_2^2=4 $$

This is a contradiction, hence $B_X=B_2$.

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