Let $\mathbb{S}^n$ be the unit sphere and choose $p_0$ as the north pole. Consider the function $d:\mathbb{S}^n \to [0, \infty)$ defined by $d(p) = d(p,p_0) = \cos^{-1}( \langle p, p_0 \rangle)$. It is the intrinsec distance to $p_0$ in the sphere. Is this function convex?
In order to answer this question, we have to show that $d \circ \gamma$ is a convex function of a real variable, for any geodesic $\gamma$ of the sphere. I showed it for geodesics issuing from $p_0$. Is it enough?