Downloads provided by UsageCountsTriangulations and a Discrete Brunn–Minkowski Inequality in the Plane
Copyright policy ) Böröczky, Károly J.; Matolcsi, Máté; Ruzsa, Imre Z.; Santos Leal, Francisco; Serra Albó, Oriol;
Triangulations and a Discrete Brunn–Minkowski Inequality in the Plane
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.
30 pages
QA Mathematics / matematika, Brunn-Minkowski theory, Geometria dels, triangulations, Convex sets in \(2\) dimensions (including convex curves), Geometry of numbers, Classificació AMS::11 Number theory::11H Geometry of numbers, Nombres, Geometria dels, :11 Number theory::11H Geometry of numbers [Classificació AMS], Nombres, Brunn–Minkowski theory, Inequalities and extremum problems involving convexity in convex geometry, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), :Matemàtiques i estadística::Àlgebra::Teoria de nombres [Àrees temàtiques de la UPC], Triangulations, Minkowski sum, Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombres
QA Mathematics / matematika, Brunn-Minkowski theory, Geometria dels, triangulations, Convex sets in \(2\) dimensions (including convex curves), Geometry of numbers, Classificació AMS::11 Number theory::11H Geometry of numbers, Nombres, Geometria dels, :11 Number theory::11H Geometry of numbers [Classificació AMS], Nombres, Brunn–Minkowski theory, Inequalities and extremum problems involving convexity in convex geometry, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), :Matemàtiques i estadística::Àlgebra::Teoria de nombres [Àrees temàtiques de la UPC], Triangulations, Minkowski sum, Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombres
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