Proofs on Unit Sphere Packings

2013 ◽  
pp. 17-56
Author(s):  
Károly Bezdek
Keyword(s):  
Unit Sphere ◽  
Sphere Packings

scholarly journals Upper bounds for packings of spheres of several radii

2014 ◽  
Vol 2 ◽  
Author(s):  
DAVID DE LAAT ◽  
FERNANDO MÁRIO DE OLIVEIRA FILHO ◽  
FRANK VALLENTIN
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Semidefinite Programming ◽  
Unit Sphere ◽  
Upper Bound ◽  
Convex Bodies ◽  
Classical Problem ◽  
Upper Bounds ◽  
Sphere Packings ◽  
Spherical Caps

AbstractWe give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve the bounds for the classical problem of packing identical spheres.

Isoperimetric Inequalities and the Dodecahedral Conjecture

1997 ◽  
Vol 08 (06) ◽  
pp. 759-780 ◽  
Author(s):  
Károly Bezdek
Keyword(s):  
Unit Sphere ◽  
Sphere Packing ◽  
Isoperimetric Inequalities ◽  
Sphere Packings ◽  
Voronoi Polyhedron ◽  
Regular Dodecahedron ◽  
Nonoverlapping Unit

The dodecahedrad conjecture, posed more than 50 years ago, says that the volume of any Voronoi polyhedron of a unit sphere packing in [Formula: see text] is at least as large as the volume of a regular dodecahedron of inradius 1. In this paper we show how the dodecahedral conjecture can be obtained from the distance conjecture of 14 and 15 nonoverlapping unit spheres and from the isoperimetric conjecture of Voronoi faces of unit sphere packings.

scholarly journals Contact graphs of unit sphere packings revisited

2013 ◽  
Vol 104 (1) ◽  
pp. 57-83 ◽  
Author(s):  
Károly Bezdek ◽  
Samuel Reid
Keyword(s):  
Unit Sphere ◽  
Sphere Packings ◽  
Contact Graphs

scholarly journals On contact numbers of totally separable unit sphere packings

2016 ◽  
Vol 339 (2) ◽  
pp. 668-676 ◽  
Author(s):  
Károly Bezdek ◽  
Balázs Szalkai ◽  
István Szalkai
Keyword(s):  
Unit Sphere ◽  
Sphere Packings

scholarly journals ON CONTACT NUMBERS OF LOCALLY SEPARABLE UNIT SPHERE PACKINGS

Mathematika ◽  
2021 ◽  
Vol 67 (3) ◽  
pp. 714-729
Author(s):  
Károly Bezdek
Keyword(s):  
Unit Sphere ◽  
Sphere Packings

scholarly journals Generalised Kochen–Specker Theorem in Three Dimensions

2021 ◽  
Vol 51 (3) ◽  
Author(s):  
Václav Voráček ◽  
Mirko Navara
Keyword(s):  
Unit Sphere ◽  
Hidden Variables ◽  
Three Dimensions ◽  
Quantum Theories

AbstractWe show that there is no non-constant assignment of zeros and ones to points of a unit sphere in $$\mathbb{R}^3$$ R 3 such that for every three pairwisely orthogonal vectors, an odd number of them is assigned 1. This is a new strengthening of the Bell–Kochen–Specker theorem, which proves the non-existence of hidden variables in quantum theories.

scholarly journals Diverse densest binary sphere packings and phase diagram

2021 ◽  
Vol 103 (2) ◽  
Author(s):  
Ryotaro Koshoji ◽  
Mitsuaki Kawamura ◽  
Masahiro Fukuda ◽  
Taisuke Ozaki
Keyword(s):  
Phase Diagram ◽  
Sphere Packings
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