Metric inequalities for polygons

Let A1, A2, . . . , An be the vertices of a polygon with unit perimeter, that is ∑n i=1 |AiAi+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 Braß 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. [5], 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.