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 ${\rm tr}(A)$ the number of triangles in any triangulation of $A$; that is, ${\rm tr}(A) = 2i+b-2$ where $b$ and $i$ are the numbers of points of $A$ in the boundary and the interior of $[A]$ (we use $[A]$ to denote "convex hull of $A$"). We conjecture the following analogue of the Brunn-Minkowski inequality: for any two point sets $A,B \subset {\mathbb R}^2$ one has \[ {\rm tr}(A+B)^{\frac12}\geq {\rm tr}(A)^{\frac12}+{\rm tr}(B)^{\frac12}. \] We prove this conjecture in several cases: if $[A]=[B]$, if $B=A\cup\{b\}$, if $|B|=3$, or if none of $A$ or $B$ has interior points.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2018
- DOI:
- 10.48550/arXiv.1812.04117
- arXiv:
- arXiv:1812.04117
- Bibcode:
- 2018arXiv181204117B
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 30 pages