MMN-2485

# A smaller cover for closed unit curves

*Wacharin Wichiramala*;

## 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.

Vol. 19 (2018), No. 1, pp. 691-698