Groupe d’études et de recherche en analyse des décisions

Maximal perimeter and maximal width of a convex small polygon

Christian Bingane

A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $$n=2^s$$ sides are unknown when $$s \ge 4$$. In this paper, we construct a family of convex small $$n$$-gons, $$n=2^s$$ with $$s\ge 4$$, and show that their perimeters and their widths are within $$O(1/n^8)$$ and $$O(1/n^5)$$ of the maximal perimeter and the maximal width, respectively. From this result, it follows that Mossinghoff's conjecture on the diameter graph of a convex small $$2^s$$-gon with maximal perimeter is not true when $$s \ge 4$$.

, 12 pages