Fits and Covers

Drapeability

“…If the minimum width t of an arc in F is large, one would expect the arc to be more or less rotund and consequently drapeable. An upper bound for K 3 is provided by the thickness b 0 ≈ 0.43893 of the broadworm, the unique arc in F whose thickness is as large as possible (see, for example, [24] and the references cited there). A lower bound is the thickness 0.25851 of the non-drapeable arc ζ 1 described above.…”

“…For an overview of such problems, see, for example, [24]. Finch [8] includes a survey of some related questions, and Finch and Wetzel [9] consider an interesting, closely related question.…”

Besicovitch triangles extended

“…These triangles were called Besicovitch triangles or Besicovitch worm covers. Wetzel [7] conjectured that the right isosceles triangle (α = π 4 ) with the hypothenuse equal to √ 1 + 1/9 is a worm cover. Indeed, in this article we show that the class of Besicovitch triangles can be extended to the isosceles triangles T α with…”

“…As forty years old problem of Leo Moser [5]: Find the convex set of least area that contains a congruent copy of each unit arc in the plane, remains unapproachable, see [7], we study a less general problem: Given a convex set K, find the smallest region similar to K that contains a congruent copy of each unit arc (worm). This region is called a worm cover.…”

Smallest universal covers for families of triangles

“…The problem is interesting both for open and closed curves, and both versions are still open. The survey by Wetzel [9] and the book by Brass et al [4] list these and other results related to universal covers.…”