Group for Research in Decision Analysis

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

## Christian Bingane

A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $$n=2^s$$ 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, we formulate the maximal perimeter problem as a nonconvex quadratically constrained quadratic optimization problem and, for $$n=2^s$$ with $$3 \le s\le 7$$, we provide near-global optimal solutions obtained with a sequential convex optimization approach.

, 18 pages