  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 $s \ge 4$. In this paper, we construct a family of convex small $n$-gons,
$n=2^s$ and $s\ge 3$, 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 $3 \le s\le 7$, we provide global optimal solutions.
