Let $(X,\|\cdot\|)$ be a real normed space. For any points $A$ and $B$ in $X$, let $AB:=\|A-B\|$. Suppose that for any points $A$ and $B$ in $X$ and any straight line $\ell\subseteq X$ such that $B\in\ell$ we have $$AB^2\ge AA_\ell^2+BA_\ell^2 $$ for some point $A_\ell$ on $\ell$ that is at the smallest distance to $A$ among all points on $\ell$ (it is easy to see that such a point $A_\ell$ always exists). Does it then necessarily follow that the norm $\|\cdot\|$ is induced by an inner product?

Perhaps easier to answer is the version of this question with
**"for some point $A_\ell$ on $\ell$ that is at the smallest distance to $A$"**
replaced by
**"for all points $A_\ell$ on $\ell$ that are at the smallest distance to $A$"**.

(I was not sure if this question is appropriate for MO, but then saw this MO question, which seems to be of the same flavor.)

theclosest to $A$ among all points on $\ell$" do you just mean "a point on $\ell$ whose distance to $A$ is minimal" or "a unique point whose distance to $A$ is minimal"? $\endgroup$4more comments