Let $(V,\|\cdot\|)$ be a real normed linear space. $V$ has the property that given any nonempty convex, closed subset $K$, there exists a unique $v_0\in K$ such that $\|v_0\| \leq \|v\|, \forall v\in K$. Does this imply $(V,\|\cdot\|)$ is a Banach Space?
(The answer is affirmative in the case when the norm comes from an inner product, which can be found here https://math.stackexchange.com/q/4815114/1075374. The question was moved here after no answers on mathstackexchange)
