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Triangulations and a Discrete Brunn–Minkowski Inequality in the Plane

  • Branko Grünbaum Memorial Issue
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Abstract

For a set A of points in the plane, not all collinear, we denote by \({\mathrm{tr}}(A)\) the number of triangles in a triangulation of A, that is, \({\mathrm{tr}}(A)=2i+b-2\), 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 \(A,B\subset {{\mathbb {R}}}^2\) one has

$$\begin{aligned} {\mathrm{tr}}(A+B)\ge {\mathrm{tr}}(A)^{1/2}+{\mathrm{tr}}(B)^{1/2}. \end{aligned}$$

We prove this conjecture in the cases where \([A]=[B]\), \(B=A\cup \{b\}\), \(|B|=3\) and if A and B have no interior points. A generalization to larger dimensions is also discussed.

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Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, 1053, Budapest, Hungary

    Károly J. Böröczky, Máté Matolcsi & Imre Z. Ruzsa

  2. Department of Mathematics, Central European University, Nador u. 9, 1051, Budapest, Hungary

    Károly J. Böröczky

  3. Budapest University of Technology and Economics, Egry J. u. 1., 1111, Budapest, Hungary

    Máté Matolcsi

  4. Depto. de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39012, Santander, Spain

    Francisco Santos

  5. Department of Mathematics, Universitat Politècnica de Catalunya, and Barcelona Graduate School of Mathematics, Barcelona, Spain

    Oriol Serra

Authors
  1. Károly J. Böröczky
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  2. Máté Matolcsi
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  3. Imre Z. Ruzsa
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  4. Francisco Santos
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  5. Oriol Serra
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Correspondence to Francisco Santos.

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Editor in Charge: Kenneth Clarkson

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Dedicated to the memory of Branko Grünbaum.

K. J. Böröczky was supported by NKFIH grants 116451, 121649 and 129630. M. Matolcsi was supported by NKFIH grant 109789. I. Z. Ruzsa was supported by NKFIH grant 109789. F. Santos was supported by grant MTM2017-83750-P of the Spanish Ministry of Science and grant EVF-2015-230 of the Einstein Foundation Berlin. O. Serra was supported by grants MTM2017-82166-P and MDM-2014-0445 of the Spanish Ministry of Science.

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Böröczky, K.J., Matolcsi, M., Ruzsa, I.Z. et al. Triangulations and a Discrete Brunn–Minkowski Inequality in the Plane. Discrete Comput Geom 64, 396–426 (2020). https://doi.org/10.1007/s00454-019-00131-9

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  • DOI: https://doi.org/10.1007/s00454-019-00131-9

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