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

For finite subsets A and B of 2, we write A + B = {a + b : a A,b 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)1 2 tr(A)1 2 + tr(B)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.

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additive combinatorics, sumsets, Brunn–Minkowski inequality, triangulations
Mathematical Subject Classification 2010
Primary: 11B75, 52C05
Received: 28 December 2012
Revised: 31 May 2013
Accepted: 22 September 2013
Published: 10 December 2014

Communicated by Andrew Granville
Károly J. Böröczky
Alfréd Rényi Institute of Mathematics
Hungarian Academy of Sciences
Reáltanoda utca 13-15
Budapest 1053
Central European University
1051 Budapest, Nádor utca 9
Benjamin Hoffman
Department of Mathematical Sciences
Lewis & Clark College
615 Palatine Hill Road
Portland, OR 97219
United States