Universal convex covering problems under translation 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 translation and discrete rotations. In particular, we show that the solution is an equilateral triangle of height 1 when translation and discrete rotation of $\pi$ are allowed. Our proof is purely geometric and elementary. We also give convex coverings of closed curves of length 2 under translation and discrete rotations of multiples of $\pi /2$ and $2\pi /3$. We show a minimality of the covering for discrete rotation of multiples of $\pi /2$, which is an equilateral triangle of height smaller than 1, and conjecture that the covering is the smallest-area convex covering. Finally, we give the smallest-area convex coverings of all unit segments under translation and discrete rotations $2\pi /k$ for all integers $k\ge 3$.