A smaller cover for closed unit curves

01/20/2018

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by   Wacharin Wichiramala, et al.

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Forty years ago Schaer and Wetzel showed that a 1/π×1/2π√(π^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 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.