Minimisation problem $||p-x|| \le ||p-v|| \forall v\in V$

Question

I am studying for an exam and dont have solutions for this exercise:

Let $p \in R^n$ and $V \subset R^n$ a subspace. Look at the following minimisation problem

$||p-x|| \le ||p-v|| \forall v\in V$.

Is $x$ unique, if the norm is convex?

What I know so far is that a norm is convex if $B(0,1) = (x\in R^n : ||x||\le1)$ is convex.

Answer 1

No. Look at the norm $\|x\| = \max_{1 \le i \le n} |x_i|$. The unit ball is convex, but consider $p = (0,0,\ldots,1)$ and $V = (x_1,\ldots,x_{n-1},0)$. There is not a unique closest point to $p$ in $V$.