I have a Banach space geometry question (a curiosity-driven spin-off from a research topic). Given a point $x$ on the unit sphere of a Banach space and a vector $y\ne 0$, there is a multiple $t_0y$ of $y$ for which $\|t_0y-x\|$ is minimized (this will be unique if the norm is strictly convex).
My question is this:
For which Banach spaces $X$ is it guaranteed that $\|t_0y\|\le \|x\|$?
My "Euclidean intuition" suggested that this should be the case for all Banach spaces, but a little experimentation showed that this is not the case. You quickly see this is really a question about two dimensions. In fact it seems to fail for every $\ell^p$, $p\ne 2$ (see the attached figure in $p=1.2$).
Could it be true that this property characterizes Hilbert space? (I looked at the obvious sources: (MO 11192 and papers mentioned in there and didn't find anything of the sort).