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Mutually contiguous translates of a plane disk. (English) Zbl 0829.52008

The authors consider topological disks (i.e. homeomorphic to the unit circle) in the euclidean plane \(E^2\). Two disks are called contiguous, if they have a common point but do not overlap. Their main result is: If \(n\) translates of a disk are mutually contiguous, then \(n \leq 4\). As the authors point out, this answers a question posed by J. Mitchell in 1991. Further several related results are given.
Reviewer: J.M.Wills (Siegen)

MSC:

52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)
05B40 Combinatorial aspects of packing and covering
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[1] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1988. · Zbl 0634.52002
[2] G. Fejes Tóth, New results in the theory of packing and covering , Convexity and its applications eds. P. M. Gruber and J. M. Wills, Birkhäuser, Basel, 1983, pp. 318-359. · Zbl 0533.52007
[3] H. Groemer, Abschätzungen für die Anzahl der konvexen Körper, die einen konvexen Körper berühren , Monatsh. Math. 65 (1961), 74-81. · Zbl 0096.16701 · doi:10.1007/BF01322659
[4] B. Grünbaum, On a conjecture of H. Hadwiger , Pacific J. Math. 11 (1961), 215-219. · Zbl 0131.20003 · doi:10.2140/pjm.1961.11.215
[5] B. Grünbaum and G. C. Shephard, Tilings and patterns , W. H. Freeman and Company, New York, 1987. · Zbl 0601.05001
[6] H. Hadwiger, Über Treffanzahlen bei translationsgleichen Eikörpern , Arch. Math. 8 (1957), 212-213. · Zbl 0080.15501 · doi:10.1007/BF01899995
[7] C. Halberg, E. Levin, and E. Straus, On contiguous congruent sets in Euclidean space , Proc. Amer. Math. Soc. 10 (1959), 335-344. JSTOR: · Zbl 0098.35801 · doi:10.2307/2032843
[8] K. Kuperberg and W. Kuperberg, Translates of a starlike plane region with a common point , Intuitive geometry (Szeged, 1991) eds. K. Böröczky and G. Fejes, Colloq. Math. Soc. János Bolyai, vol. 63, North-Holland, Amsterdam, 1994, pp. 205-216. · Zbl 0823.52005
[9] H. Minkowski, Dichteste gitterförmige Lagerung kongruenter Körper , Nachr. Ges. Wiss. Göttingen (1904), 311-355. · JFM 35.0508.02
[10] J. R. Munkres, Elements of algebraic topology , Addison-Wesley Publishing Company, Menlo Park, CA, 1984. · Zbl 0673.55001
[11] C. A. Rogers, Packing and covering , Cambridge Tracts in Mathematics and Mathematical Physics, No. 54, Cambridge University Press, New York, 1964. · Zbl 0176.51401
[12] H. Voderberg, Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente , Jahresber. Deutsch. Math.-Verein 46 (1936), 229-231. · Zbl 0015.31502
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