A smaller cover for closed unit curves

Abstract

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.keywords: covering of unit arcs, covering by convex sets, worm problem MSC code: 52A38

Errata and Addenda to Mathematical Constants

“…New bounds 0.096694 < µ ′ < 0.112126 appear in [999]. Relevant progress is described in [1000,1001,1002,1003]. We mention, in Figure 8.3, that the quantity x = sec(ϕ) = 1.0435901095... is algebraic of degree six [1004,1005]:…”

A convex cover for closed unit curves has area at least 0.1

“…Also, they modified this pentagon to a curvilinear hexagon which has an area of less than 0.11213. In 2018, Wichiramala [18] showed that the opposite corner of the pentagon can be clipped to be a hexagonal cover with area less than 0.11023 which is the best currently known bound.…”