Let $(M^n,g)$ be a smooth Riemannian manifold with non-empty boundary $\partial M$. Assume (for simplicity) that $M$ is compact. Let $M$ be locally geodesically convex, i.e. any shortest path in $M$ connecting any two sufficiently close points is a geodesic in $M$ (namely satisfies the standard second order ODE).
Assume in addition that $M$ has non-negative sectional curvature. Is it true that the function on $M$ given by $$x\mapsto dist(x,\partial M)$$ is concave, i.e. its restriction on any shortest path in concave?
A reference would be helpful.
