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 $v_2$


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