Tight bounds on the maximal perimeter of convex equilateral small polygons

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 >= 4. In this work, we construct a family of convex equilateral small n-gons, for n = 2(s) and s >= 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.