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 $\mathrm{tr}(A)$ the number of triangles in any triangulation of $A$; that is, $\mathrm{tr}(A)=2i+b-2$ 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.

30 pages

References in corpus (1)

• A generalization of the discrete version of Minkowski's fundamental theorem