On smoothness of quasihyperbolic balls

TL;DR

This paper proves that quasihyperbolic balls are $C^1$-smooth in uniformly smooth Banach spaces and possibly in Euclidean spaces, exploring their properties, geodesics, and applications to Banach space renormings.

Contribution

It establishes the $C^1$-smoothness of quasihyperbolic balls in uniformly smooth Banach spaces, a result likely new even in Euclidean spaces, and explores their applications.

Findings

Quasihyperbolic balls are $C^1$-smooth in uniformly smooth Banach spaces.

The smoothness of quasihyperbolic balls is addressed in Euclidean spaces, possibly for the first time.

Application to Banach space renormings using quasihyperbolic metrics.

Abstract

We investigate properties of quasihyperbolic balls and geodesics in Euclidean and Banach spaces. Our main result is that in uniformly smooth Banach spaces a quasihyperbolic ball of a convex domain is $C^1$-smooth. The question about the smoothness of quasihyperbolic balls is old, originating back to the discussions of F.W. Gehring and M. Vuorinen in 1970's. To our belief, the result is new also in the Euclidean setting. We also address some other issues involving the smoothness of quasihyperbolic balls. We introduce an interesting application of quasihyperbolic metrics to renormings of Banach spaces. To provide a useful tool for this approach we turn our attention to the variational stability of quasihyperbolic geodesics. Several examples and illustrations are provided.