Besicovitch triangles cover unit arcs

Abstract

A long standing conjecture is that the Besicovitch triangle, i.e., an equilateral triangle with side √ 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.

Besicovitch triangles extended

“…An arc inside of a triangle T is circumscribed by T if it touches all sides of T. The arcs of interest resemble a staple or letters s, w. Staple arcs, or w-arcs satisfying (4), are those in [2], s-arcs will satisfy (2) instead of (3) below.…”

“…Proof We place the arc as a -arc (C on top, B, D on the x-axis) in T α -left standard position. In every possible configuration listed in Theorem 4.1 of [2] there is a shorter staple-arc, or a w-arc satisfying (4), or an s-arc satisfying (3). Since our s-arcs satisfy a more restricted requirement (2), the configurations involving s-arcs must be re-examined.…”

“…The conjecture was proven in [2] in search of isosceles triangular worm covers, where it was shown that each isosceles triangle T α , having the base angle α ≥ arctan 5 3 and the base b as in (1) below, covers unit arcs. When α ≤ π 3 , triangles T α are the smallest covers in their similarity classes.…”

“…As in [2] we consider only simple polygonal arcs referring to the result of [4]: If a convex set covers any simple polygonal unit arc it covers any unit arc. The idea of the proof is the same as in [2].…”

“…The idea of the proof is the same as in [2]. We build a small pool of "long" arcs greater than worms, then any arc, placed in so-called standard -position between the bases of T α and its upsidedown copy, that escapes from both triangles must be visibly larger than an arc from this pool.…”