A note on triangulations of sumsets

Abstract

For finite subsets $A$ and $B$ of ${ℝ}^2$, we write $A+B=\left\{a+b:a\in A,b\in B\right\}$. We write $tr\left(A\right)$ to denote the common number of triangles in any triangulation of the convex hull of $A$ using the points of $A$ as vertices. We consider the conjecture that $tr{\left(A+B\right)}^{\frac{1}{2}}\ge tr{\left(A\right)}^{\frac{1}{2}}+tr{\left(B\right)}^{\frac{1}{2}}$. If true, this conjecture would be a discrete two-dimensional analogue to the Brunn–Minkowski inequality. We prove the conjecture in three special cases.