A smaller cover for closed unit curves

Abstract

Forty years ago Schaer and Wetzel showed that a $\frac{1}{\pi }\times \frac{1}{2\pi }\sqrt{{\pi }^2-4}$ rectangle, whose area is about $0.12274,$ is the smallest rectangle that is a cover for the family of all closed unit arcs. More recently Füredi and Wetzel showed that one corner of this rectangle can be clipped to form a pentagonal cover having area $0.11224$ for this family of curves. Here we show that then the opposite corner can be clipped to form a hexagonal cover of area less than $0.11023$ for this same family. This irregular hexagon is the smallest cover currently known for this family of arcs.

In the appendix, the computer code for numerical optimization is provided together with explanation. The link to the actual file, a Mathematica notebook, is at www.math.sc.chula.ac.th/~wacharin/optimization/closed%20arcs