Complex Geodesics on Convex Domains

TL;DR

This paper investigates the existence and uniqueness of complex geodesics in convex domains within Banach spaces, providing new conditions and formulas, especially for classical spaces like nd ll^p.

Contribution

It establishes new existence and uniqueness results for complex geodesics in convex Banach domains, including explicit formulas for ll^p spaces.

Findings

Existence of complex geodesics in the unit ball of certain Banach spaces.

Uniqueness of geodesics in strictly convex domains with the Radon-Nikodym property.

Explicit formulas for geodesics in ll^p spaces.

Abstract

Existence and uniqueness of complex geodesics joining two points of a convex bounded domain in a Banach space $X$ are considered. Existence is proved for the unit ball of $X$ under the assumption that $X$ is 1-complemented in its double dual. Another existence result for taut domains is also proved. Uniqueness is proved for strictly convex bounded domains in spaces with the analytic Radon-Nikodym property. If the unit ball of $X$ has a modulus of complex uniform convexity with power type decay at 0, then all complex geodesics in the unit ball satisfy a Lipschitz condition. The results are applied to classical Banach spaces and to give a formula describing all complex geodesics in the unit ball of the sequence spaces $\ell^p$ ($1 \leq p < \infty$).