A smaller cover for closed unit curves

TL;DR

This paper improves the known minimal-area cover for all closed unit arcs by introducing a smaller irregular hexagon, surpassing previous rectangular and pentagonal covers in efficiency.

Contribution

It presents the first known irregular hexagonal cover with smaller area than prior rectangular and pentagonal covers for closed unit arcs.

Findings

The irregular hexagon has an area less than 0.11023.

It is the smallest known cover for the family of all closed unit arcs.

The hexagon improves upon previous covers in area efficiency.

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 $0.122\,74,$ is the smallest rectangle that is a cover for the family of all closed unit arcs. More recently F\"{u}redi 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.