Metric inequalities for polygons

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.Comment: 13 pages, 2 figures. This version replaces the previous version from 8 Feb 2011. A new section has been added and the material has been reorganized; a correction has been done in the proof of Lemma 4 (analysis of Case 3