Polygons and non-starlikeness of Teichmuller spaces

The problem of starlikeness of Teichmuller spaces in Bers' embedding was raised in 1974 and is solved (negatively) for Teichmuller spaces of sufficiently large dimensions. The original proof given by the author relies on the existence of conformally rigid domains established by Thurston. Later the author found another proof of non-starlikeness of universal Teichmuller space based on geometric features of rectilinear polygons. This paper provides a complete solution of the problem for Teichmuller spaces $T(g,0)$ of closed Riemann surfaces of genus $g \ge 2$.