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Failure of averaging on multiply-connected domains. (English) Zbl 0726.30034
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.

30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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