Tight bounds on the maximal perimeter of convex equilateral small polygons

Abstract

A small polygon is a polygon that has diameter one. The maximal perimeter of a convex equilateral small polygon with $$n=2^s$$ sides is not known when $$s \ge 4$$ . In this work, we construct a family of convex equilateral small n-gons, for $$n=2^s$$ and $$s \ge 4$$ , and show that their perimeters are within $$O(1/n^4)$$ of the maximal perimeter and exceed the previously best known values from the literature. In particular, for the first open case $$n=16$$ , our result proves that Mossinghoff’s equilateral hexadecagon is suboptimal.