Abstract :

In 1964, Toth showed that among the spherical polygons of equal area having at most n sides the regular n-gon has the least possible perimeter and stated that this result on spherical polygons can be extended without difficulty to hyperbolic polygons. In 1984, Bezdek stated that among all n-gons of fixed perimeter in the hyperbolic plane, the regular n-gon has the largest area. We do not find any proof of this result in the literature. In this talk, we present a proof of this isoperimetric inequality in the hyperbolic plane. We also present a complete proof of this isoperimetric inequality for spherical case.