Besicovitch triangles cover unit arcs

Abstract

A long standing conjecture is that the Besicovitch triangle, i.e., an equilateral triangle with side ${\sqrt{28/ 27},}$ is a worm-cover. We will show that indeed there exists a class of isosceles triangles, that includes the above equilateral triangle, where each triangle from the class is a worm-cover. These triangles are defined so that the shortest symmetric z-arc stretched from side to side and touching the base would have length one.