Observations on quasihyperbolic geometry modeled on Banach spaces

TL;DR

This paper investigates the quasihyperbolic metric in Banach spaces, providing criteria for geodesic smoothness and addressing open questions about quasihyperbolic balls, advancing understanding of geometric properties in infinite-dimensional spaces.

Contribution

It introduces a new criterion for geodesic smoothness under a Dini condition and answers an open question on the smoothness of quasihyperbolic balls in Banach spaces.

Findings

Established a smoothness criterion for quasihyperbolic geodesics

Provided conditions under which quasihyperbolic balls are smooth

Improved previous results on quasihyperbolic geometry in Banach spaces

Abstract

In this paper, we continue our study of quasihyperbolic metric in Banach spaces. The main results of the paper present a criterion for smoothness of geodesics of quasihyperbolic type metrics in Banach spaces, under a Dini type condition on the weight function, which improves an earlier result of the two first authors. We also answer to a question posed by the two first authors in an earlier paper with R. Kl\'en, and present results related to the question on smoothness of quasihyperbolic balls.