Given two constant-speed geodesics $\gamma_1$ and $\gamma_2$ in an euclidean space $\mathbb E^n$, it is possibile to see that: $$ t \mapsto d(\gamma_1(t), \gamma_2(t)) $$ is a convex function.
The same statement is false, even locally, in the $2$--sphere, as it is easy to see. Using (maybe) CAT($\kappa$) considerations, I strongly suspect that the following inequality should be true in every Riemannian manifold:
Theorem(?): Let $M$ be a riemannian manifold, $\epsilon > 0$ and let $p \in M$. Then, there exists a convex neighorhood $U = U(M, p, \varepsilon)$ of $p$ in $M$ such that, if $\gamma_1$ and $\gamma_2 :\, [0, 1] \to U$ are constant-speed geodesics contained in $U$, then: $$ d(\gamma_1(t), \gamma_2(t)) \le (1+\varepsilon) \max\{d(\gamma_1(0), \gamma_2(0)), d(\gamma_1(1), \gamma_2(1))\} $$ for every $t \in [0, 1]$, where $d$ is the Riemannian distance on $M$.
This result would holds for $\varepsilon = 0$ in when $ t \mapsto d(\gamma_1(t), \gamma_2(t)) $ is convex.