Covering families of triangles

Abstract

A cover for a family $${{\mathcal {F}}}$$ F of sets in the plane is a set into which every set in $${{\mathcal {F}}}$$ F can be isometrically moved. We are interested in the convex cover of smallest area for a given family of triangles. Park and Cheong conjectured that any family of triangles of bounded diameter has a smallest convex cover that is itself a triangle. The conjecture is equivalent to the claim that for every convex set $${{\mathcal {X}}}$$ X there is a triangle Z whose area is not larger than the area of $${{\mathcal {X}}}$$ X , such that Z covers the family of triangles contained in $${{\mathcal {X}}}$$ X . We prove this claim for the case where a diameter of $${{\mathcal {X}}}$$ X lies on its boundary. We also give a complete characterization of the smallest convex cover for the family of triangles contained in a half-disk, and for the family of triangles contained in a square. In both cases, this cover is a triangle.