TITLE: The projection of a point onto a convex set is unique with respect to any norm
QUESTION [6 upvotes]: Given a convex set $C$, the projection of a point $z$ onto $C$ is a point $x$ in $C$ that minimizes $\|z- x\|$. Say the minimum is achieved at $x^*\in C$. My textbook shows such $x^*$ is unique under the euclidean norm, as shown below. My guess is the uniqueness should hold regardless what norm is chosen, but I have trouble proving it. Thanks!

REPLY [3 votes]: Counterexample: Take $\mathbb R^2$ with $\|\cdot\|_1$. Be $C=\{x\in\mathbb R^2:\|x\|_1\le1\}$. Be $P=(2,2)$. Then for any point $x_\lambda=(\lambda,1-\lambda)$ with $\lambda\in[0,1]$ you have $\|P-x_\lambda\|_1 = |2-\lambda|+|2-1+\lambda| = 3$, which is also the minimum distance.
Note that $\|\cdot\|_1$ is not a differentiable function, therefore the proof doesn't apply.