#### Vol. 8, No. 1, 2015

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A note on triangulations of sumsets

### Károly J. Böröczky and Benjamin Hoffman

Vol. 8 (2015), No. 1, 75–85
##### 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.

##### Keywords
additive combinatorics, sumsets, Brunn–Minkowski inequality, triangulations
##### Mathematical Subject Classification 2010
Primary: 11B75, 52C05