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Some extremal problems on the hyperbolic polygons

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 [6].