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Petersburg Department of Steklov Institute of Mathematics

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PREPRINT 11/1998

A. Yu. SOLYNIN
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SOME EXTREMAL PROBLEMS
ON THE HYPERBOLIC POLYGONS

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.

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