Sausages are good packings
type: Item
instance of: scholarly article
title: Sausages are good packings
zbMATH Open document ID: 0829.52010
DOI: 10.1007/BF02574046
author: Martin Henk
author: Ulrich. Betke
author: Jörg M. Wills
published in: Discrete \& Computational Geometry
publication date: 1995-07-02T00:00:00Z
full work available at URL: https://eudml.org/doc/131364
review text: In a previous paper [J. Reine Angew. Math. 453, 165-191 (1994; Zbl 0797.52010)] the authors introduced a new original approach to the study of finite and infinite packings and were able to prove several important results (e.g. they confirmed Fejes Tóth's ``sausage'' conjecture in high dimensions). Their approach is primarily based on the following parametric density: Let \(C_n\) be a set of \(n\) points defining a packing arrangement of the unit ball \(B^d\) in \(d\)-dimensional Euclidean space; then, for \(\rho > 0\) the density \(\delta (B^d, C_n, \rho)\) of the packing is defined to be the ratio of \(n\) times the volume of \(B^d\) over the volume of the body \(\text{conv } C_n + \rho B^d\). In the present paper, the authors prove that, for \(\rho < \sqrt {2}\) and sufficiently high dimensions, ``sausage'' packings of balls (that is packings for which \(C_n\) is contained in a line segment) minimize the density \(\delta (B^d, C_n, \rho)\). This gives an improvement of the bound \(\rho < \sqrt {3}\) proved in the quoted article. The above density can easily be generalized to arbitrary convex bodies, and in this case the bound \(\sqrt {2}\) has to be replaced by a constant depending on the dimension of the space and on the inradius and circumradius of the given body. Moreover, it is shown how results for infinite packings can be deduced by this density.
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 11H31
zbMATH DE Number: 764228
zbMATH Keywords: finite packings
zbMATH Keywords: sausage arrangements
zbMATH Keywords: density
reviewed by: C. Peri
MaRDI profile type: Publication
cites work: Slices of L. Fejes Tóth's sausage conjecture
cites work: Q4301219
cites work: Q4023809
cites work: Q4274020
cites work: Q3751649
cites work: Über die Einlagerung von Kreisen in einen konvexen Bereich
cites work: Existenzsätze für Lagerungen im Euklidischen Raum
cites work: Q4274021
cites work: Q5791470
cites work: Q4194048
cites work: An inequality in the geometry of numbers
cites work: The closest packing of convex two-dimensional domains
cites work: Q5568974
OpenAlex ID: W2077855085
Recommended article: Finite packings of spheres
Recommended article: Kugelpackungen- Altes und Neues
Recommended article: Finite sphere packings and critical radii
Recommended article: A new approach to covering
Recommended article: Radii and the Sausage Conjecture
Recommended article: The sausage conjecture holds for convex hulls of moderately bent sausages
Recommended article: Q3750548
Recommended article: Packings, sausages and catastrophes
Recommended article: Mean projections and finite packings of convex bodies
description: scientific article; zbMATH DE number 764228