A note on triangulations of sumsets

Böröczky, Károly; Hoffman, Benjamin

doi:10.2140/involve.2015.8.75

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A note on triangulations of sumsets

Károly J. Böröczky and Benjamin Hoffman

Vol. 8 (2015), No. 1, 75–85

DOI: 10.2140/involve.2015.8.75

Abstract

For finite subsets $A$ and $B$ of $ℝ^{2}$ , we write $A + B = {a + b : a \in A, b \in B}$ . We write $tr (A)$ 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 $tr {(A + B)}^{\frac{1}{2}} \geq tr {(A)}^{\frac{1}{2}} + tr {(B)}^{\frac{1}{2}}$ . 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

Milestones

Received: 28 December 2012

Revised: 31 May 2013

Accepted: 22 September 2013

Published: 10 December 2014

Communicated by Andrew Granville

Authors

Károly J. Böröczky
Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Reáltanoda utca 13-15 Budapest 1053 Hungary	Central European University 1051 Budapest, Nádor utca 9 Hungary

Benjamin Hoffman
Department of Mathematical Sciences Lewis & Clark College 615 Palatine Hill Road Portland, OR 97219 United States

Vol. 8, No. 1, 2015