Perimeter Generating Functions For The Mean-Squared Radius Of Gyration Of Convex Polygons

TL;DR

This paper derives series expansions for the perimeter generating functions related to the radius of gyration of convex polygons, revealing exact solutions and confirming a size exponent of 1 across cases.

Contribution

It introduces new series expansions and exact solutions for the perimeter generating functions of convex polygons' radius of gyration.

Findings

Size exponent $ u=1$ for all cases

Derived algebraic exact solutions

Numerical series expansions obtained

Abstract

We have derived long series expansions for the perimeter generating functions of the radius of gyration of various polygons with a convexity constraint. Using the series we numerically find simple (algebraic) exact solutions for the generating functions. In all cases the size exponent $\nu=1$.