Is every closed convex subset a sublevel set?

Question

Is it true that on every Riemannian manifold $M$ (whether compact or merely complete), every closed convex set C in M is the sublevel set $f((-\infty,t])$ of some convex function $f : M \rightarrow \mathbb{R}$? Thank you every much!

Answer 1

Copying my pseudo-comments into the answer box:

1. The equator of $\mathbb S^2$ is convex but is not a sublevel set of any convex function: otherwise the function would have a maximum somewhere on the sphere.

2. But if $M$ is a $CAT(0)$ space, then the distance function to any closed convex subset of $M$ is convex. See p.178 of Metric spaces of nonpositive curvature by Bridson and Haefliger.