On complex convexity

Abstract: This thesis is about complex convexity. We compare it with other notions of convexity such as ordinary convexity, linear convexity, hyperconvexity and pseudoconvexity. We also do detailed study about ?-convex Hartogs domains, which leads to a definition of ?-convex functions of class C1. The study of Hartogs domains also leads to characterization theorem of bounded ?-convex domains with C1 boundary that satisfies the interior ball condition. Both the method and the theorem is quite analogous with the known characterization of bounded ?-convex domains with C2 boundary. We also show an exhaustion theorem for bounded ?-convex domains with C2 boundary. This theorem is later applied, giving a generalization of a theorem of L. Lempert concerning the relation between the Carathéodory and Kobayashi metrics.