Maximal perimeter and maximal width of a convex small polygon

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 $s \ge 4$\mathrm{.}Inthispaper,weproposeanapproachtoconstructconvexsmall$n$-gonsoflargeperimeterandlargewidthwhen$n=2^s$with$s\ge 2$. Assuming the existence of an axis of symmetry, a convex small $n$-gon is described as a composition of $n\mathrm{/}2$ and both its perimeter and its width are given as functions of a single variable. By selecting the composition that minimizes the violation of a cycle constraint by a particular solution, the $n$-gons constructed outperform the best $n$-gons found in the literature. For example, for $n=64$, the perimeter and the width obtained are within ${10}^{-22}$ and ${10}^{-12}$ of the maximal perimeter and the maximal width, respectively. From our results, it appears that Mossinghoff’s conjecture on the diameter graph of a convex small $2^s$-gon with maximal perimeter is not true when $s \ge 4$.