Suppose that $V$ is a finite-dimensional real vector space equipped with some $\ell_p$ norm for $1 < p < \infty$. ($p$ is chosen this way so as to be strictly convex.)
Then suppose we have an affine subspace $S \subset V$, and we want to find the shortest vector $v \in S$ such that $||v||_p$ is minimized.
Now suppose we also have a second affine subspace $R \subset S \subset V$. We again want to find the shortest vector $v_2 \in R$ such that $||v_2||_p$ is minimized.
Does $v_2$ also minimize the distance $||v_2-v||_p$?
That is, does $v_2$ minimize not only the $\ell_p$ distance to the origin, but also the distance to $v$, the shortest vector in the larger subspace?
This seems to hold for $p=2$. Does it hold for all $\ell_p$ norms? For all strictly convex norms?