Polynomial hulls with convex sections and interpolating spaces

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