# Geodesic convexity on Riemannian manifolds with boundary

Let $M$ be a smooth, connected and bounded manifold and let $f:M \rightarrow \mathbb{R}$ be a smooth function.

The conditions to ensure that $f$ is a geodesically convex on $M$ is to ensure firstly that $M$ is geodesically convex manifold. To do that Hopf-Rinow theorem states that $M$ must be a complete topological space. Secondly, the covariant derivative of $f$, $D^2 f$ is non-negative (or equivalently its geodesic hessian matrix $H_g f$ is semi-definite positive).

Assume that $D^2 f$ is non-negative on the topological closure of $M$.

My question is : Is $f$ a geodesically convex function on $\overline{M}$?

• First, there is no reason for $\bar M$ to still be a Riemannian manifold; metric completion loses smoothness most of the time. Second, even if $\bar M$ happens to be smooth, there is no reason for the extension of $f$ to $\bar M$ to be so. In particular, the symbol $D^2 f$ might make no sense. Third, in order to extend $f$ to a continuous function on $\bar M$ you will have to assume it uniformly continuous; assuming $f$ to have compact support seems reasonable. In any case, you will have to make so many assumptions on $M$ and $f$, that the problem becomes irrelevant. – Alex M. Jul 5 at 9:54