Smallest universal covers for families of triangles

Abstract

A universal cover for a family T of triangles is a convex set that contains a congruent copy of each triangle T is an element of T. We conjecture that for any family T of triangles of bounded diameter there is a triangle that forms a universal cover for T of smallest possible area. We prove this conjecture for all families of two triangles, and for the family of triangles that fit in the unit disk.