Groupe d’études et de recherche en analyse des décisions

# Tight bounds on the maximal perimeter of convex equilateral small polygons

## Christian Bingane et Charles Audet

A small polygon is a polygon of unit diameter. The maximal perimeter of a convex equilateral small polygon with $$n=2^s$$ vertices is not known when $$s \ge 4$$. In this paper, we construct a family of convex equilateral small $$n$$-gons, $$n=2^s$$ and $$s\ge 4$$, and show that their perimeters are within $$\pi^4/n^4 + O(1/n^5)$$ of the maximal perimeter and exceed the previously best known values from the literature. For the specific cases where $$n=32$$ and $$n=64$$, we present solutions whose perimeters are even larger, as they are within $$1.1 \times 10^{-5}$$ and $$2.1 \times 10^{-6}$$ of the optimal value, respectively.

, 12 pages