On the symplectic volume of the moduli space of spherical and Euclidean polygons

Abstract

In this paper, we study the symplectic volume of the moduli space of polygons by using Witten's formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed side-lengths. The main result of our is that among all the polygons with fixed perimeter in S³ or E³ the regular one is the most flexible and that among all the spherical polygons the regular one with side-length π/2 is the most flexible.