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arXiv · 1509.00546

A characterization of cut locus for $C^1$ hypersurfaces

Tatsuya Miura
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Abstract

Let $\Omega$ be an open set in $\mathbb{R}^n$ with $C^1$-boundary and $\Sigma$ be the skeleton of $\Omega$, which consists of points where the distance function to $\partial\Omega$ is not differentiable. This paper characterizes the cut locus (ridge) $\overline{\Sigma}$, which is the closure of the skeleton, by introducing a generalized radius of curvature and its lower semicontinuous envelope. As an application we give a sufficient condition for vanishing of the Lebesgue measure of $\overline{\Sigma}$.

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Tatsuya Miura. 2015-09-02. A characterization of cut locus for $C^1$ hypersurfaces. https://doi.org/10.1007/s00030-016-0413-y

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