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


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.


Download Citation

Minoru TANAKA. "The Singular Locus of an Almost Distance Function." Tokyo J. Math. 43 (1) 47 - 74, June 2020.


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

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

Primary: 53C22
Secondary: 53C60

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


This article is only available to subscribers.
It is not available for individual sale.

Vol.43 • No. 1 • June 2020
Back to Top