Compact sets and the closure of their convex hulls in CAT(0) spaces

TL;DR

This paper investigates the properties of convex hulls of compact sets in CAT(0) spaces, introduces a new operation called threading, and explores implications for the computability of the Fréchet mean.

Contribution

It introduces the threading operation for sets in CAT(0) spaces and proves its properties, including preservation of compactness and its role in convex hull closure.

Findings

In flat complete CAT(0) spaces, the convex hull of a compact set has a compact closure.

Threading operation exhibits monotonicity and preserves compactness.

The Fréchet mean of finite sets is computable in finitely many steps in finite type spaces.

Abstract

We study the closure of the convex hull of a compact set in a complete CAT(0) space. First we give characterization results in terms of compact sets and the closure of their convex hulls for locally compact CAT(0) spaces that are either regular or satisfy the geodesic extension property. Later inspired by a geometric interpretation of Carath\'eodory's Theorem we introduce the operation of threading for a given set. We show that threading exhibits certain monotonicity properties with respect to intersection and union of sets. Moreover threading preserves compactness. Next from the commutativity of threading with any isometry mapping we prove that in a flat complete CAT(0) space the closure of the convex hull of a compact set is compact. We apply our theory to the computability of the Fr\'echet mean of a finite set of points and show that it is constructible in at most a finite number of steps, whenever the underlying space is of finite type.