Shortest path through random points

Abstract

Let $(M,g_{1})$ be a complete $d$-dimensional Riemannian manifold for $d>1$. Let $\mathcal{X}_{n}$ be a set of $n$ sample points in $M$ drawn randomly from a smooth Lebesgue density $f$ supported in $M$. Let $x,y$ be two points in $M$. We prove that the normalized length of the power-weighted shortest path between $x,y$ through $\mathcal{X}_{n}$ converges almost surely to a constant multiple of the Riemannian distance between $x,y$ under the metric tensor $g_{p}=f^{2(1-p)/d}g_{1}$, where $p>1$ is the power parameter.