A smaller cover for closed unit curves

Wacharin Wichiramala;


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

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