Ruzsa–Matolcsi conjecture on triangulations of planar sum sets

Let $A$ and $B$ be finite non-collinear point sets in $\mathbb{R}^2$. For a finite non-collinear set $S$, let $b_S$ and $i_S$ denote the numbers of points of $S$ on the boundary and in the interior of its convex hull, respectively. Since the number of triangles in a triangulation of the convex hull is $\operatorname{tr}(S)=b_S+2i_S-2$, Ruzsa–Matolcsi's conjecture.

$$\sqrt{2i_{A+B}+b_{A+B}-2}\geq \sqrt{2i_A+b_A-2}+\sqrt{2i_B+b_B-2}.$$

Equivalently, $\operatorname{tr}(A+B)^{\frac12}\geq \operatorname{tr}(A)^{\frac12}+\operatorname{tr}(B)^{\frac12}$. This is a planar discrete analogue of the Brunn–Minkowski inequality; the paper verifies it in special cases, while the general conjecture remains open.