Andreas
Potschka
TU Clausthal
On small convex polygons with maximum diameter
Abstract.
A polygon in the plane is called small if its diameter is at most 1. We consider the problem of finding a small convex polygon of maximum perimeter for a given number of vertices. Karl Reinhardt proved in 1922 that such maximal polygons need to be equilateral if the number of vertices has an odd divisor. If the number of vertices is a power of two, the problem has been settled only for the quadrilateral and the octagon. We propose a new geometric construction based on zonogons (axisymmetric polygons), which leads to a Mixed-Integer Nonlinear Program approach, and provide a two phase algorithm for computing polygons with large perimeter numerically to high precision. Based on this approach and convergence estimates for a Lagrange-Newton-type method, we derive a class of polygons, for which we prove a superalgebraic asymptotic bound on the gap of their perimeter to an upper bound due to Reinhardt.