# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Densest packing of translates of the union of two circles. (English) Zbl 0606.52004
Let $d(u)$ (resp. $\bar d(u))$ denote the density of the densest packing (resp. lattice packing) of translates of the domain u. An open conjecture is that if u is the union of two convex domains having a point in common, then $d(u)=\bar d(u)$. The author proves this in the case where u is the union of two unit-radius circular discs whose centers have distance at most 2. The paper closes with two conjectures --- one for lattice packings of translates of unions of unit discs, the other for packings of congruent copies of a domain.
Reviewer: W.Moser
##### MSC:
 52C17 Packing and covering in $n$ dimensions (discrete geometry) 52A40 Inequalities and extremum problems (convex geometry)
##### Keywords:
density; lattice packings of translates
Full Text:
##### References:
 [1] C. A. Rogers, The closest packing of convex two-dimensional domains.Acta Math. 86 (1951), 309--321. · Zbl 0044.19203 · doi:10.1007/BF02392671 [2] L. Fejes Tóth, Some packing and covering theorems.Acta Sci. Math. (Szeged) 12/A (1950), 62--67. · Zbl 0037.22102 [3] L. Fejes Tóth, On the densest packing of convex discs.Mathematika 30 (1983), 1--3. · Zbl 0523.52010 · doi:10.1112/S0025579300010354 [4] L. Fejes Tóth, Densest packing of translates of a domain.Acta Math. Acad. Sci. Hungar. 45 (1984), 437--440. · Zbl 0574.52014 [5] L. Fejes Tóth, Über einen geometrischen Satz.Math. Z. 46 (1940), 83--85. · Zbl 66.0902.03 · doi:10.1007/BF01181430 [6] J. M. Wills, Research Problem 35.Period. Math. Hungar. 14 (1983), 312--314.