Complex Geodesics on Convex Domains

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 l (1 ≤ p < ∞). In this article, we discuss the existence, uniqueness and continuity of complex geodesics on a convex domain D in a complex Banach space X . The term ‘complex geodesic’ is due to Vesentini [33], although the concept was discussed by Carathéodory [5] and Reiffen [27] under the name ‘metric plane’. Recent results on this topic are to be found in [11, 14, 15, 16, 34, 35, 36, 37]. Applications of complex geodesics to the study of biholomorphic automorphisms and to fixed point sets are to be found in [5, 33, 34, 35, 36, 37]. Our results on the existence problem depend on topological properties of the Banach space X , the results on uniqueness depend on the geometry of the boundary ∂D and on an analytic-geometric property of X (the analytic Radon-Nikodym property), while the continuity (i.e. continuous extensions to the boundary) is obtained using complex uniform convexity. In section 1, we introduce complex geodesics and related concepts and prove some basic