Let $X$ be a nice compact subset of $R^d$. Given a function $p: X \to R^+$, define the length of a path $\gamma \subset X$ as $\ell(\gamma) = \int_\gamma p(x) dx$, and the distance between two points $x,y \in X$ as the length of the shortest path connecting them. I'd like to prove that if $p$ is "nice" (say, continuously differentiable), then the shortest paths are smooth and have a bounded curvature. Is there an elementary way of proving this, without diving too deep into Riemannian geometry? I'd in particular like to understand what are the minimal assumptions on $p$ I need.

Thanks!

Let $X$ be a nice compact subset of $R^d$. Given a function $p: X \to \mathbb{R}^+$, define the length of a path $\gamma \subset X$ as $\ell(\gamma) = \int_\gamma p \, ds$, and the distance between two points $x,y \in X$ as the length of the shortest path connecting them. I'd like to prove that if $p$ is "nice" (say, continuously differentiable), then the shortest paths are smooth and have a bounded curvature. Is there an elementary way of proving this, without diving too deep into Riemannian geometry? I'd in particular like to understand what are the minimal assumptions on $p$ I need.