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On separability in discrete geometry

Károly Bezdek, Zsolt Lángi

TL;DR

The article surveys separability in discrete geometry, focusing on non-separable arrangements and totally separable packings across Euclidean and spherical spaces. It leverages NS- and TS-packing frameworks to present minimal-covering results, density bounds, and contact-number analyses, drawing on foundational work by Erdős, Fejes Tóth, Goodman–Goodman, and Lutwak, among others, while highlighting key open problems and dimensional extensions. Core contributions include precise covering bounds for NS-families, density characterizations of λ- and ρ-separable packings across geometries, crystallization-type results for locally separable packings, and high-dimensional bounds linking packing structure to hull geometry. Collectively, the survey consolidates methods and results that bridge classical geometric inequalities with modern discrete-geometry packings, outlining significant open directions in higher dimensions and curved spaces.

Abstract

A problem of Erdős (Amer. Math. Monthly 52: 494-498, 1945) and a theorem of Fejes Tóth and Fejes Tóth (Acta Math. Acad. Sci. Hungar. 24: 229-232, 1973) initiated the study of non-separable arrangements of convex bodies and the investigation of totally separable packings of convex bodies with both topics analyzing the concept of separability from the point view of discrete geometry. This article surveys the progress made on these and some closely related problems and highlights the relevant questions that have been left open.

On separability in discrete geometry

TL;DR

The article surveys separability in discrete geometry, focusing on non-separable arrangements and totally separable packings across Euclidean and spherical spaces. It leverages NS- and TS-packing frameworks to present minimal-covering results, density bounds, and contact-number analyses, drawing on foundational work by Erdős, Fejes Tóth, Goodman–Goodman, and Lutwak, among others, while highlighting key open problems and dimensional extensions. Core contributions include precise covering bounds for NS-families, density characterizations of λ- and ρ-separable packings across geometries, crystallization-type results for locally separable packings, and high-dimensional bounds linking packing structure to hull geometry. Collectively, the survey consolidates methods and results that bridge classical geometric inequalities with modern discrete-geometry packings, outlining significant open directions in higher dimensions and curved spaces.

Abstract

A problem of Erdős (Amer. Math. Monthly 52: 494-498, 1945) and a theorem of Fejes Tóth and Fejes Tóth (Acta Math. Acad. Sci. Hungar. 24: 229-232, 1973) initiated the study of non-separable arrangements of convex bodies and the investigation of totally separable packings of convex bodies with both topics analyzing the concept of separability from the point view of discrete geometry. This article surveys the progress made on these and some closely related problems and highlights the relevant questions that have been left open.
Paper Structure (14 sections, 35 theorems, 56 equations, 10 figures)

This paper contains 14 sections, 35 theorems, 56 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Minimal coverings of non-separable arrangements
  3. Non-separable arrangements in Euclidean spaces
  4. More on non-separable translative families of convex bodies
  5. Non-separable arrangements in spherical spaces
  6. Dense totally separable packings
  7. Finding the densest totally separable (finite or infinite) translative packings in the Euclidean plane
  8. Finding the densest totally separable packings of a given number of congruent spherical caps in the spherical plane
  9. On densest $\lambda$-separable circle packings in planes of constant curvature
  10. On densest separable translative packings in Euclidean spaces of dimensions greater than $\mathbf{2}$
  11. On contact numbers for separable translative packings
  12. On contact numbers for totally separable translative packings
  13. Largest contact numbers and crystallization for locally separable unit disk packings
  14. Upper bounding the contact numbers of locally separable unit sphere packings in dimensions larger than $\mathbf{2}$

Key Result

Theorem 2.1

Let the disks $\mathbf B[\mathbf x_1, \tau_1]\subset {\mathbb E}^2, \dots , \mathbf B[\mathbf x_n,\tau_n]\subset {\mathbb E}^2$ have the following property: No line of ${\mathbb E}^2$ divides the disks $\mathbf B[\mathbf x_1, \tau_1], \dots , \mathbf B[\mathbf x_n,\tau_n]$ into two non-empty familie

Figures (10)

  • Figure 1: A counterexample in the plane for three triangles.
  • Figure 2: A counterexample in ${\mathbb R}^3$ for three tetrahedra. The dotted lines denote the intersection of a plane touching all three tetrahedra with the smallest tetrahedron containing them.
  • Figure 3: An example for equality in (\ref{['thm:areaformula']}.1).
  • Figure 4: TS-packings by translates of a triangle and a unit disk for which equality is attained in (\ref{['thm:areaformula']}.2) in Theorem \ref{['thm:areaformula']}.
  • Figure 5: A TS-packing of translates of $\mathbf K$ (with $\mathbf K$ being a circular disk for the sake of simplicity), which satisfies the conditions in Theorem \ref{['thm:Oler']} and for which there is equality in (\ref{['eq:Oler']}) of Theorem \ref{['thm:Oler']}.
  • ...and 5 more figures

Theorems & Definitions (84)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Definition 1
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • Corollary 2.7
  • ...and 74 more