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June 2020 The Singular Locus of an Almost Distance Function
Minoru TANAKA
Tokyo J. Math. 43(1): 47-74 (June 2020). DOI: 10.3836/tjm/1502179298

Abstract

The aim of this article is to generalize the notion of the cut locus and to get the structure theorem for it. For this purpose, we first introduce a class of 1-Lipschitz functions on a Finsler manifold, each member of which is called an {\em almost distance function}. Typical examples of an almost distance function are the distance function from a point and the Busemann functions. The generalized notion of the cut locus in this paper is called the {\em singular locus} of an almost distance function. The singular locus consists of the {\em upper singular locus} and the {\em lower singular locus}. The upper singular locus coincides with the cut locus of a point $p$ for the distance function from the point $p$, and the lower singular locus coincides with the set of all copoints of a ray $\gamma$ when the almost distance function is the Busemann function of the ray $\gamma$. Therefore, it is possible to treat the cut locus of a closed subset and the set of copoints of a ray in a unified way by introducing the singular locus for the almost distance function. In this article, some theorems on the distance function from a closed set and the Busemann function are generalized by making use of the almost distance function.

Citation

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Minoru TANAKA. "The Singular Locus of an Almost Distance Function." Tokyo J. Math. 43 (1) 47 - 74, June 2020. https://doi.org/10.3836/tjm/1502179298

Information

Published: June 2020
First available in Project Euclid: 24 August 2019

zbMATH: 07227181
MathSciNet: MR4121789
Digital Object Identifier: 10.3836/tjm/1502179298

Subjects:
Primary: 53C22
Secondary: 53C60

Rights: Copyright © 2020 Publication Committee for the Tokyo Journal of Mathematics

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Vol.43 • No. 1 • June 2020
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