<p>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 ℂ-convex Hartogs domains, which leads to a definition of ℂ-convex functions of class <i>C</i>¹. The study of Hartogs domains also leads to characterization theorem of bounded ℂ-convex domains with <i>C</i>¹ boundary that satisfies the interior ball condition. Both the method and the theorem is quite analogous with the known characterization of bounded ℂ-convex domains with <i>C</i>² boundary. We also show an exhaustion theorem for bounded ℂ-convex domains with <i>C</i>² boundary. This theorem is later applied, giving a generalization of a theorem of L. Lempert concerning the relation between the Carathéodory and Kobayashi metrics.</p>

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 ℂ-convex Hartogs domains, which leads to a definition of ℂ-convex functions of class C1. The study of Hartogs domains also leads to characterization theorem of bounded ℂ-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 ℂ-convex domains with C2 boundary. We also show an exhaustion theorem for bounded ℂ-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.