The Singular Locus of an Almost Distance Function

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.