Conical geodesic bicombings on subsets of normed vector spaces

TL;DR

This paper investigates the existence and uniqueness of conical geodesic bicombings in subsets of normed spaces, providing new examples and conditions under which these bicombings are linear and unique.

Contribution

It introduces the first example of a non-consistent convex geodesic bicombing and establishes conditions for linearity and uniqueness in normed spaces.

Findings

Existence of a non-consistent convex geodesic bicombing.

Under mild conditions, bicombings on open subsets are linear.

Unique consistent bicombing exists for convex subsets with interior.

Abstract

We prove existence and uniqueness results for conical geodesic bicombings on subsets of normed vector spaces. Concerning existence, we give a first example of a non-consistent convex geodesic bicombing. Furthermore, we show that under a mild geometric assumption on the norm, a conical bicombing on an open subset of a normed space locally consists of linear geodesics. As an application, we obtain by the use of a Cartan-Hadamard type result that if a closed convex subset of a Banach space has non-empty interior, then it admits a unique consistent conical geodesic bicombing, namely the one given by linear segments.