On Quasihyperbolic Geodesics in Banach Spaces

TL;DR

This paper investigates the properties of quasihyperbolic geodesics in Banach spaces, demonstrating conditions for uniqueness, existence, and smoothness, thereby advancing understanding of geometric structures in infinite-dimensional analysis.

Contribution

It establishes new results on the uniqueness, existence, and smoothness of quasihyperbolic geodesics in various classes of Banach spaces, including strictly convex and uniformly convex spaces.

Findings

Quasihyperbolic geodesics are unique in strictly convex Banach spaces with the Radon-Nikodym property.

There exist convex domains in Banach spaces with no geodesics between certain points.

In uniformly convex Banach spaces with power-type modulus of convexity, all geodesics are smooth.

Abstract

We study properties of quasihyperbolic geodesics on Banach spaces. For example, we show that in a strictly convex Banach space with the Radon-Nikodym property, the quasihyperbolic geodesics are unique. We also give an example of a convex domain $\Omega$ in a Banach space such that there is no geodesic between any given pair of points $x, y \in \Omega\,.$ In addition, we prove that if $\mathrm{X}$ is a uniformly convex Banach space and its modulus of convexity is of a power type, then every geodesic of the quasihyperbolic metric, defined on a proper subdomain of $\mathrm{X}$, is smooth.