The Singular Locus of an Almost Distance Function

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, each member of which is called an {\it almost distance function}. Typical examples of an almost distance function are the distance function from a point and the Busemann function on a complete Riemannian manifold. The generalized notion of the cut locus in this paper is called the {\it singular locus} of an almost distance function. The singular locus consists of the upper one and the lower one. The upper singular locus coincides with the cut locus of a point for the distance function from the point, and the lower singular locus coincides with the set of all copoints of a ray when the almost distance function is the Busman function of the ray. 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, we obtain the structure theorem (Theorem B) for the singular locus of an almost distance function on a 2-dimensional Finsler manifold that contains both structure theorems ([S], Theorem B) and [Sa,Theorem 2.13]) for the cut locus and the set of copoints of a ray as a corollary.