Tight bounds on the maximal perimeter and the maximal width of convex small polygons

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) vertices are not known when s >= 4. In this paper, we construct a family of convex small n-gons, n = 2(s) and s >= 3, and show that the perimeters and the widths obtained cannot be improved for large n by more than a/n(6) and b/n(4) respectively, for certain positive constants a and b. In addition, assuming that a conjecture of Mossinghoff is true, we formulate the maximal perimeter problem as a nonlinear optimization problem involving trigonometric functions and, for n = 2(s) with 3 <= s <= 7, we provide global optimal solutions.

Automated summary

This paper investigates the maximal perimeter and maximal width problem of convex small polygons, and presents a family of convex small n-gons where the perimeters and widths cannot be improved for large n. The paper also formulates the maximal perimeter problem as a nonlinear optimization problem involving trigonometric functions, and provides global optimal solutions for specific cases of n = 2(s) with 3 <= s <= 7.