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Triangulations and a discrete Brunn-Minkowski inequality in the plane

Abstract: For a set A of points in the plane, not all collinear, we denote by tr(A) the number of triangles in a triangulation of A, that is, tr(A) = 2i + b - 2, where b and i are the numbers of boundary and interior points of the convex hull [A] of A respectively. We conjecture the following discrete analog of the Brunn?Minkowski inequality: for any two finite point sets A, B c R2 one has tr(A + B) >= tr(A)1/2 + tr(B)1/2. We prove this conjecture in the cases where [A]=[B], B = A U {b}, |B| = 3 and if A and B have no interior points. A generalization to larger dimensions is also discussed.

 Autoría: Böröczky K.J., Matolcsi M., Ruzsa I.Z., Santos F., Serra O.,

 Fuente: Discrete and Computational Geometry, 2020, 64(2), 396-426

 Editorial: Springer New York LLC

 Fecha de publicación: 29/08/2020

 Nº de páginas: 31

 Tipo de publicación: Artículo de Revista

 DOI: 10.1007/s00454-019-00131-9

 ISSN: 0179-5376,1432-0444

 Proyecto español: MTM2017-83750-P

 Url de la publicación: http://doi.org/10.1007/s00454-019-00131-9

Autoría

BÖRÖCZKY, KÁROLY J.

MATOLCSI, MÁTÉ

RUZSA, IMRE Z.

SERRA, ORIOL