Lineally convex domains of finite type: holomorphic support functions

Abstract

Smooth bounded lineally convex domains of finite type constitute a natural class of domains in complex analysis, since they are locally biholomorphically invariant. A smooth family of holomorphic support functions is constructed by an almost explicit formula on every such domain. It satisfies the best possible estimates near the point of support on every two-dimensional transverse affine intersection with the domain. Together with a suitable pseudometric on these domains, it will allow to do precise quantitative complex analysis by integral kernels on them.