  A small polygon is a polygon of unit diameter. The maximal width of an
equilateral small polygon with $n=2^s$ vertices is not known when $s \ge 3$.
This paper solves the first open case and finds the optimal equilateral small
octagon. Its width is approximately $3.24\%$ larger than the width of the
regular octagon: $\cos(\pi/8)$. In addition, the paper proposes a family of
equilateral small $n$-gons, for $n=2^s$ with $s\ge 4$, whose widths are within
$O(1/n^4)$ of the maximal width.

Metric Geometry;Combinatorics;Optimization and Control