Sectorial Covers for Unit Arcs

摘要

It has been conjectured in conjunction with Leo Moser's “worm” problem that a circular sector with angle 30? and radius 1 contains a congruent copy of each unit arc in the plane. If this is true, this sector would be the smallest such set currently known. This conjecture remains unsettled. We show that for 0?< ? < 90? the circular sector with vertex angle ? and radius r = csc 2? contains a congruent copy of each unit arc. For ? = 30?, this means that the radius r = 2/3√3 ≈ 1.1548 suffices. Our arguments depend on an interesting extension to angles of a result of Coulton and Movshovich about parallel support lines of an arc.