On complex convexity

by Jacquet, David

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.