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 $||v^\star||\leq ||x||$?