Let $X$ be a finite-dimensional normed space, and $V\subset X$ a vector subspace. For a point $x\in X$ we define a closest projection of $x$ onto $V$ to be a minimizer $v^\star$ of $\|v-x\|$ amongst $v\in V$.

Under what conditions on the norm, can we guarantee that for all $V$ and $x$ there exist a projection $v^\star$ of $x$ onto $V$ for which furthermore there holds $\|v^\star\|\leq \|x\|$?

note based on comments It's obvious that $\|v^\star - x\|\leq \|x\|$, but I didn't ask that. Euclidean norm satisfies the property and max-norm on $\mathbb R^2$ doesn't satisfy it, as fedja points out in a comment. The question is, what norms do/don't satisfy?

Please if you put a downvote, kindly explain why, so that I understand my own errors?