### 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) = 2 i+ 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⊂ R^{2} 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∪ { b} ,

Original language | English |
---|---|

Journal | Discrete and Computational Geometry |

DOIs | |

Publication status | Accepted/In press - Jan 1 2019 |

### Fingerprint

### Keywords

- Brunn–Minkowski theory
- Minkowski sum
- Triangulations

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Discrete and Computational Geometry*. https://doi.org/10.1007/s00454-019-00131-9

**Triangulations and a Discrete Brunn–Minkowski Inequality in the Plane.** / Böröczky, Károly J.; Matolcsi, M.; Ruzsa, I.; Santos, Francisco; Serra, Oriol.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*. https://doi.org/10.1007/s00454-019-00131-9

}

TY - JOUR

T1 - Triangulations and a Discrete Brunn–Minkowski Inequality in the Plane

AU - Böröczky, Károly J.

AU - Matolcsi, M.

AU - Ruzsa, I.

AU - Santos, Francisco

AU - Serra, Oriol

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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) = 2 i+ 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⊂ 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∪ { b} ,

AB - 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) = 2 i+ 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⊂ 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∪ { b} ,

KW - Brunn–Minkowski theory

KW - Minkowski sum

KW - Triangulations

UR - http://www.scopus.com/inward/record.url?scp=85071432547&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85071432547&partnerID=8YFLogxK

U2 - 10.1007/s00454-019-00131-9

DO - 10.1007/s00454-019-00131-9

M3 - Article

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

ER -