Shortest Path through Random Points

Details Shortest Path through Random Points Shortest Path through Random Points

2012-01-31

arxiv.org/abs/1202.0045v1

Mathematics

Probability

Scientific paper

Let $(M,g_1)$ be a compact $d$-dimensional Riemannian manifold. Let $\mscrX_n$ be a set of $n$ sample points in $M$ drawn randomly from a Lebesgue density $f$. Define two points $x,y \in M$. We prove that the normalized length of the power-weighted shortest path between $x,y$ passing through $\mscrX_n$ converges to the Riemannian distance between $x,y$ under the metric $g_p = f^{2(1-p)/d} g_1$, where $p > 1$ is the power parameter. The result is an extension of the Beardwood-Halton-Hammersley theorem to shortest paths.

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