# Triangulations and a discrete Brunn-Minkowski inequality in the plane

## TL;DR

This paper explores a discrete analogue of the Brunn-Minkowski inequality in the plane, relating the number of triangles in triangulations of point sets and proving the conjecture in specific cases.

## Contribution

The paper introduces a conjecture extending the Brunn-Minkowski inequality to triangulation counts and proves it in several special scenarios.

## Key findings

- Proved the conjecture when sets have the same convex hull.
- Established the inequality for sets where one is a union with a single point.
- Confirmed the inequality for sets with no interior points.

## 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.