no. 2
Failure of averaging on multiply connected domains
Annales de l'Institut Fourier, Tome 40 (1990) no. 2, pp. 357-370.

Nous démontrons que pour toute surface de Riemann ouverte X, de groupe fondamental non-abélien, il existe une fonction analytique multiforme f sur X telle que l’enveloppe convexe par fibre du graphe de f ne contienne le graphe d’aucune fonction holomorphe uniforme sur X.

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.

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     author = {Barrett, David E.},
     title = {Failure of averaging on multiply connected domains},
     journal = {Annales de l'Institut Fourier},
     pages = {357--370},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {40},
     number = {2},
     year = {1990},
     doi = {10.5802/aif.1216},
     zbl = {0726.30034},
     mrnumber = {92b:30050},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1216/}
}
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Barrett, David E. Failure of averaging on multiply connected domains. Annales de l'Institut Fourier, Tome 40 (1990) no. 2, pp. 357-370. doi : 10.5802/aif.1216. https://aif.centre-mersenne.org/articles/10.5802/aif.1216/

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