Covering families of triangles

A cover for a family $\mathcal{F}$ of sets in the plane is a set into which every set in $\mathcal{F}$ 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 $\mathcal{X}$ there is a triangle $Z$ whose area is not larger than the area of $\mathcal{X}$, such that $Z$ covers the family of triangles contained in $\mathcal{X}$. We prove this claim for the case where a diameter of~$\mathcal{X}$ 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.