# Shortest Path through Random Points

@article{Hwang2012ShortestPT,
title={Shortest Path through Random Points},
author={Sung Jin Hwang and Steven Benjamin Damelin and Alfred O. Hero},
journal={arXiv: Probability},
year={2012}
}
• Published 31 January 2012
• Mathematics
• arXiv: Probability
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…
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