# Minimizing $\ell_p$ distance to affine subspace of affine subspace

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?

It does not hold for all $$\ell_p$$ norms. Somehow, I have got the feeling that it holds only for $$\ell_2$$, but I have no proof for that.
We can construct a counterexample for $$\ell_p$$ where $$p$$ is close to one. Since the shape of the unit ball for $$p\to 1^+$$ is very close to that in $$\ell_1$$ we may think in $$\ell_1$$ sense first and construct a counterexample that is robust to small variations in parameters. Let's take $$V={\Bbb R}^3$$ and $$S\colon 3x+2y=3,\qquad R\colon\begin{cases}3x+2y=3,\\ 3z+2y=3.\end{cases}$$ Then minimizing $$\|v\|_1$$ for $$v\in S$$ we get $$v=(1,0,0)$$, and $$\|v_2\|_1$$ for $$v_2\in R$$ we get $$v_2=(0,\frac32,0)$$. However, $$\|v_2-v\|_1=\frac52$$ is not the shortest as $$\|v_3-v\|_1=1$$ for $$v_3=(1,0,1)$$.
We now argue that the optimal points $$v$$, $$v_2$$ and $$v_3$$ remain optimal for small variations of parameters (coefficients) in $$S$$ and $$R$$, then it is expected that the optimal points are going to be continuous in $$p$$ when $$p\to 1^+$$. Indeed, the numerical minimization for $$p=1.1$$ confirms that the optimal points are close to those above. I've tried Wolfram Alpha
Optimal $$v$$
Optimal $$v_2$$