Polynomial hulls with convex sections and interpolating spaces

Abstract

Assume that L ⊂ ∂ D × C m L \subset \partial D \times {{\mathbf {C}}^m} is compact and has convex vertical sections. Denote by K K its polynomially convex hull. It is shown that K ∖ ∂ D × C m K\backslash \partial D \times {{\mathbf {C}}^m} , if nonempty, can be covered by graphs of analytic functions f : D → C m f:D \to {{\mathbf {C}}^m} . The proof is based on complex interpolation theory for families of finite-dimensional normed spaces.