Failure of averaging on multiply-connected domains (Q802797)

We show that for every open Riemann surface X with non-abelian fundamental group there is a multiple-valued function f on X such that the fiberwise convex hull of the graph of f fails to contain the graph of a single-valued holomorphic function on X. This follows in part from the fact that X does not admit a homotopically-trivial antiholomorphic self- map.