A simple proof of the isoperimetric theorem for the hyperbolic plane

A. Skopenkov

Published 2010-09-05, updated 2017-10-11Version 2

In this pedagogical note we present a short proof of the following main result of arxiv.org/abs/0911.5319, and clarify its relation to the isoperimetric problem. On the hyperbolic plane consider triangles ABC with fixed lengths of AB and AC. The maximal area of these triangles is attained for the triangle ABC such that \angle A=\angle B+\angle C. The proof is essentially the same as in the above-cited paper. However, since the result is beautiful and important, a short proof cleared of unnecessary details could be interesting for a reader. The note is accessible for students familiar with elementary hyperbolic geometry. (This note is based on referee reports to the above paper, see www.mccme.ru/mmks)