The space of convex domains in complex Euclidean space

Abstract

In this mostly expository article, we describe some properties of the space of convex domains in complex Euclidean space (endowed with the local Hausdorff topology). In particular, we give careful proofs that the Kobayashi metric, the Bergman kernel/metric, and the Kahler–Einstein metric are all continuous on the space of convex domains. The group of affine automorphisms acts on this space and we also describe the orbit closures for some special classes of domains.