The planar discrete Brunn–Minkowski conjecture for triangulation counts

For finite two-dimensional sets $A,B one$ in the plane that are not collinear, let $[A]$ and $[B]$ denote their convex hulls, and let ${\rm tr}(A)$ and ${\rm tr}(B)$ be the numbers of full-dimensional simplices in triangulations using the respective sets as vertex sets. Planar discrete Brunn–Minkowski conjecture.

$${\rm tr}(A+B)^{\frac12}\geq {\rm tr}(A)^{\frac12}+{\rm tr}(B)^{\frac12}.$$

This replaces volume in the Brunn–Minkowski inequality by the number of triangles in a triangulation; in the plane this number is independent of the triangulation. The conjecture is presented as the basic planar case of the paper’s proposed discrete analogue.