This preprint was accepted March 17, 1998
Contact: A. Yu. Solynin
ABSTRACT: We study some isoperimetric problems for plane polygons. In particular we show that among all hyperbolic $n$-gons with a fixed number of sides the regular one has the maximal value of the ratio ``conformal radius perimeter''. For $n$-gon admitting a full $n$-sides reflection by the ``amplification coefficient'' we mean the ratio of the conformal radii of given and reflected polygons. We prove that the amplification coefficient takes the minimal value only for the regular $n$-gons, that confirms the conjecture posed by J.Hersch.