Geodesic bicombings on some hyperspaces

TL;DR

This paper demonstrates how certain geodesic bicombings on a metric space can be extended to its hyperspaces, including convex, compact, and bounded subsets, especially in normed spaces and $ ext{R}$-trees.

Contribution

It introduces a method to construct conical bicombings on hyperspaces from bicombings on the original space, extending geodesic structures to complex hyperspaces.

Findings

Constructs a conical bicombing on hyperspaces of convex subsets.

Extends bicombings to hyperspaces of normed spaces and $ ext{R}$-trees.

Analyzes geodesic bicombings on compact subsets in proper metric spaces.

Abstract

We show that if $(X,d)$ is a metric space which admits a consistent convex geodesic bicombing, then we can construct a conical bicombing on $CB(X)$, the hyperspace of nonempty, closed, bounded, and convex subsets of $X$ (with the Hausdorff metric). If $X$ is a normed space or an $\mathbb{R}$-tree, this same method produces a consistent convex bicombing on $CB(X)$. We follow this by examining a geodesic bicombing on the nonempty compact subsets of $X$, assuming $X$ is a proper metric space.