This is an announcement for the paper "Complex geodesics on convex domains" by Sean Dineen and Richard M. Timoney.

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 $\ell^p$ ($1 \leq p < \infty$).

Archive classification: math.FA math.CV math.MG

Mathematics Subject Classification: 46G20; 32H15; 46B45; 53C22

Citation: Progress in Functional Analysis, North Holland Mathematical

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http://front.math.ucdavis.edu/0907.1194

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http://arXiv.org/abs/0907.1194

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