On the cut locus in Alexandrov spaces and applications to convex surfaces

Abstract

Alexandrov spaces are a large class of metric spaces that includes Hilbert spaces, Riemannian manifolds and convex surfaces. In the framework of Alexandrov spaces, we examine the ambiguous locus of analysis and the cut locus of differential geometry, proving a general bisecting property, showing how small the ambiguous locus must be, and proving that typically the ambiguous locus and a fortiori the cut locus are dense.

On typical degenerate convex surfaces

“…A recent result of Zamfirescu [39] proves, under very general hypotheses (see Lemma 14), a density property of M(K ) in a compact Alexandrov space.…”

“…Cut loci have been studied for long time in Riemannian geometry (see, for example, [17] or [20]), and in the last years have been introduced for convex surfaces or Alexandrov spaces (see, for example, [18,22,38,39]). …”

What do cylinders look like?

“…Theorem 3.1 can also be used (see §3 in [13]) to show the existence of convex surfaces S ⊂ R 3 (e.g., cylinders with 3 planar symmetries), such that S has a closed curve O most points of which are endpoints of S, and also has a simple closed geodesic crossing O. Other examples of such cut loci have been obtained in [3,6,10,13,15,17].…”

Mutually nearest and farthest points of sets and the Drop Theorem in geodesic spaces

“…A nice synthesis of issues concerning nearest and farthest point problems in connection with geometric properties of Banach spaces and some extensions of these problems can be found in [4]. Zamfirescu initiated in [24] the investigation of this kind of problems in the context of geodesic spaces. Later on, researchers have focused on adapting the ideas of Stečkin [23] into the geodesic setting.…”