Universal convex covering problems under translations and discrete rotations

Abstract

We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently, closed curves of length 2) allowing translations and discrete rotations. In particular, we show that the solution is an equilateral triangle of height 1 when translations and discrete rotations of π are allowed. We also give convex coverings of closed curves of length 2 under translations and discrete rotations of multiples of π/2 and of 2π/3. We show that no proper closed subset of that covering is a covering for discrete rotations of multiples of π/2, which is an equilateral triangle of height smaller than 1, and conjecture that such a covering is the smallest-area convex covering. Finally, we give the smallest-area convex coverings of all unit segments under translations and discrete rotations of 2π/k for all integers k=3.