Metric inequalities for polygons

TL;DR

This paper establishes tight bounds on the sum of pairwise distances among vertices of polygons with unit perimeter, extending known results to non-convex polygons, and provides simplified proofs for maximum perimeter problems in disks.

Contribution

It derives new tight estimates for pairwise distance sums in polygons, including non-convex cases, and offers a simpler proof for the maximum perimeter problem in disks.

Findings

Tight bounds for sum of pairwise distances in polygons.

Exact formulas for maximum perimeter of polygons in a disk.

Simplified proof for the maximum perimeter problem.

Abstract

Let $A_1,A_2,...,A_n$ be the vertices of a polygon with unit perimeter, that is $\sum_{i=1}^n |A_i A_{i+1}|=1$. We derive various tight estimates on the minimum and maximum values of the sum of pairwise distances, and respectively sum of pairwise squared distances among its vertices. In most cases such estimates on these sums in the literature were known only for convex polygons. In the second part, we turn to a problem of Bra\ss\ regarding the maximum perimeter of a simple $n$-gon ($n$ odd) contained in a disk of unit radius. The problem was solved by Audet et al. \cite{AHM09b}, who gave an exact formula. Here we present an alternative simpler proof of this formula. We then examine what happens if the simplicity condition is dropped, and obtain an exact formula for the maximum perimeter in this case as well.