Abstract
Let J be a system of sets. If all members of J are contained in a given set C and each point of C belongs to at most one member of J then J is said to be a packing into C. If, on the other hand, each point of C belongs to at least one member of J then we say that J is a covering of C. Packings and coverings have been considered in various spaces and on various combinatorial structures. Here we are interested in problems concerning packings and coverings consisting of convex bodies in spaces of constant curvature, i.e. in Euclidean, spherical and hyperbolic space. Instead of saying that J is a packing into the whole space or J is a covering of the whole space we shall simply use the terms J is a packing and J is a covering.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abramowitz, M. and Stegun, I.A. (1972) Handbook of mathematical functions, National Bureau of Standards Applied Math. Series 55, U.S. Dept. Commerce, Washington, D.C.
Askey, R. (1975) Orthogonal polynomials and special functions, Regional Conference Lectures in Applied Mathematics, SIAM 21.
Bambah, R.P. (1954) On lattice coverings by spheres, Proc. Nat Inst. Sci. India 20, 25–52.
Bambah, R.P. (1970) Packing and covering, Math. Student 38, 133–138.
Bambah, R.P. (1971) Geometry of numbers, packing and covering and discrete goemetry, Math. Student 39, 117–129.
Bambah, R.P., Rogers, C.A. and Zassenhaus, H. (1964) On coverings with convex domains, Acta Arith. 9, 191–207.
Bambah, R.P. and Solane, N.J. (1982) On a problem of Ryskov concerning lattice coverings, Acta Arith. 42, 107–109.
Bannai, E., Solane, N.J.A. (1981) Uniqueness of certain spherical codes, Canad. J. Math. 33, 437–449.
Baranovskii, E.P. (1964) On packing of n-dimensional Euclidean spaces by equal spheres (Russian), Izv. Vys`s. Ucebn. Zved. Matematika 39, 14–24.
Bârâny, I., Füredi, Z. and Pach, J. (1983) Discrete convex functions and proof of the six circle conjecture of Fejes Tóth (to appear).
Barnes, E.S. and Sloane, N.J.A. (1983) New lattice packings of spheres, Canad. J. Math. 35, 117–130.
Beck, A. and Blecker, M.N. (1971) Packing convex sets into a similar set, Acta Math. Acad. Sci. Hungar. 22, 283–303.
Bender, C. (1874) Bestimmung der grössten Anzahl gleich grosser Kugeln, welche sich auf eine Kugel von demselben Radius, wie die übrigen, auflegen lassen, Arch. Math. Phys. 56, 302–312.
Bender, E.A. (1962) Area-perimeter relations for two-dimensional lattices, Amer. Math. Monthly 69, 742–744.
Bérczi, Sz. and Nagy, D. (1980) Periodicity of extremal goemetric arrangements (densest packing, thinnest covering, tessellations), Acta Geol. Sci. Hungar. 23, 173–200.
Best, M.R. (1980) Binary codes with a minimum distance of four, IEEE Trans. Inform. Theory IT-26, 738–742.
Betke, U. and Wills, J.M. (1979) Stetige und diskrete Funktionale konvexer Körper, Contributions to geometry, Proc. Symp. Siegen 1978, 226–237.
Betke, U. and Gritzmann, P. (1983) Über L. Fejes Tóths Wurstvermutung in kleinen Dimensionen.
Betke, U., Gritzmann, P. and Wills, J.M. (1983) Slices of Fejes Tóth’s sausage conjecture, Mathematika (to ppear).
Bezdek, A. (1979) Solid packing of circles in the hyperbolic plane, Studia Sci. Math. Hungar. 14, 203–207.
Bezdek, A. (1980) Remark on the closest packing of convex discs, Studia Sci. Math. Hungar. 15, 283–285.
Bezdek, A. (1983a) Ausfüllung und Überdeckung der Ebene durch Kreise, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (to appear).
Bezdek, A. (1983b) Locally separable circle packings, Studia Sci. Math. Hungar. 18 (to appear).
Bezdek, A. (1983c) On the thinnest double saturated packing of equal circles, Studia Sci. Math. Hungar. 18 (to appear).
Bezdek, A. and Bezdek, K. (1983) Eine hinreichende Bedingung für die Überdeckung des Einheitswürfels durch homothetische Exemplare in dem n-dimensionalen euklidischen Raum, Beiträge zur Algebra und Geometrie (to appear).
Bezdek, K. (1977) Mosaics with each face having the same number of neighbours, Mat. Lapok 28, 297–303.
Bezdek, K. (1979) Optimal coverings of circles (Hungarian) Thesis, Budapest.
Bezdek, K. (1982a) Ausfüllung eines Kreises durch kongruente Kreise in der hyperbolischen Ebene, Studia Sci. Math. Hungar. 17 (to appear).
Bezdek, K. (1982b) Über Ionenpackungen, Studia Sci. Math. Hungar. 17 (to appear).
Bezdek, K. (1983a) Ausfüllungen in der hyperbolischen Ebene durch endliche Anzahl kongruenter Kreise, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (to appear).
Bezdek, K. (1983b) Über einige optimale Konfigurationen von Kreisen, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (to appear).
Bezdek, K. (1983c) Über einige Kreisüberdeckungen, Beiträge zur Algebra und Geometrie (to appear).
Blachman, N.M. and Few, L. (1963) Multiple packing of spherical caps, Mathematika 10, 84–88.
Bleicher, M.N. (1975) The thinnest three dimensional point lattice trapping a sphere, Studia Sci. Math. Hungar. 10, 157–170.
Blichfeldt, H.F. (1929) The minimum value of quadratic forms and the closest packing of spheres, Math. Ann. 101, 605–608.
Blichfeldt, H.F. (1934) The minimum values of positive quadratic forms in six, seven and eight variables, Math. Z. 39, 1–15.
Blind, G. (1969) Über Unterdeckungen der Ebene durch Kreise, J. reine angew. Math. 236, 145–173.
Blind, G. (1972) Zugängliche Unterdeckungen der Ebene durch kongruente Kreise, J. Reine Angew. Math. 257, 29–46.
Blind, G. (1974) Überdeckung der Ebene durch inkongruente Kreise, Math. Z. 140, 179–194.
Blind, G. (1975) Unterdeckung der Ebene durch inkongruente Kreise, Arch. Math. (Basel) 26, 441–448.
Blind, G. (1976) r-zugängliche Unterdeckungen der Ebene durch kongruente Kreise I, J. Reine Angew. Math. 288, 1–23.
Blind, G. (1977) r-zugängliche Unterdeckungen der Ebene durch kongruente Kreise II, J. Reine Angew. Math. 289, 1–29.
Blind, G. (1981) p-zugängliche Unterdeckungen der Sphäre durch kongruente Kreise, Resultate Math. 4, 141–154.
Blind, G. and Blind, R. (1978) Zugänglichkeit von Kugelpackungen im ℝn, Arch. Math. (Basel) 30, 438–439.
Blind, G. and Blind, R. (1979) r-zugängliche Unterdeckungen der Ebene durch kongruente Kreise, Studia Sci. Math. Hungar. 14
Blind, G. and Blind, R. (1982) Eine Abschätzung für die Dichte der dichtesten Packung mit Reuleaux-Dreiecken, Studia Sci. Math. Hunger. 17 (to appear).
Bloh, E.L. (1956) On the most dense arrangement of spherical segments on a hypersphere (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 20, 707–712.
Blundon, W.J. (1957) Multiple covering of the plane by circles, Mathematika 4, 7–16.
Blundon, W.J. (1963a) Multiple packing of circles in the plane, J. London Math. Soc. 38, 176–182.
Blundon, W.J. (1963b) Note on a paper of A. Heppes, A.ta Math. Acad. Sci. Hungar. 14, 317.
Blundon, W.J. (1964) Some lower bounds for density of multiple packing, Canad. Math. Bull. 7, 565–572.
Blundon, W.J. (1977) A three-fold non-lattice covering, Canad. Math. Bull. 20, 29–31.
Blundon, W.J. (1978) A nine-fold packing, Acta Math. Acad Sci. Hungar. 32, 293–294.
Bokowski, J.H., Hadwiger, H. and Wills, J.M. (1972) Eine Ungleichung zwischen Volumen, Oberfläche und Gitterpunktanzahl konvexer Körper im n-dimensionalen euklidischen Raum, Math. Z. 127, 363–364.
Bolle, U. (1976) Mehrfache Kreisanordnungen in der euklidischen Ebene, Dissertation, Universität Dortmund.
Bolle, U. (1979) Dichteabschätzungen für mehrfache gitterförmige Kugelanordnungen im R“’, Studia Sci. Math. Hungar. 14, 51–68.
Bolle, U. (1982) Dichteabschätzungen für mehrfache gitterförmige Kugelanordnungen im Rm II, Studia Sci. Math. Hungar. 18 (to appear).
Bolle, U. (1983) Über die Dichte mehrfacher, gitterförmigen Kreisanordnungen in der Ebene, Studia Sci. Math. Hungar. 18 (to appear).
Bollobâs, B. (1968) Remarks to a paper of L. Fejes Tóth, Studia Sci. Math. Hungar. 3, 373–379.
Bollobâs, B. (1972) The optimal structure of market areas, J. Economic Theory 4, 174–179.
Bollobâs, B. (1973) The optimal arrangement of producers, J. London Math. Soc. (2) 6, 605–613.
Boltjanski, W.G. and Gockberg, I.Z. (1965) Theorems and problems in combinatorial goemetry (Russian), Nauka, Moscow; German translation: Sätze und Probleme der kombinatorischen Geometrie, Berlin (1972).
Boyd, D.W. (1970a) Lower bounds for the disk-packing constant, Math. Comp. 24, 697–704.
Boyd, D.W. (1970b) Osculatory packings by spheres, Canad. Math. Bull. 13, 59–64.
Boyd, D.W. (1971a) On the exponent of an osculatory packing, Canad. J. Math. 23, 355–363.
Boyd, D.W. (1971b) The disc-packing constant, Aequationes Math. 7, 182–193.
Boyd, D.W. (1972) Disk packings which have non-extreme exponents, Canad. Math. Bull. 15, 341–344.
Boyd, D.W. (1973a) Improved bounds for the disk-packing constant, Aequationes Math. 9, 99–106.
Boyd, D.W. (1973b) An algorithm for generating the sphere coordinates in a three-dimensional osculatory packing, Math. Comp. 27, 369–377.
Boyd, D.W. (1973c) The osculatory packing of a three dimensional sphere, Canad. J. Math. 25, 303–322.
Boyd, D.W. (1973d) The residual set dimension of the Apollonian packing, Mathematika 20, 170–174.
Boyd, D.W. (1974) A new class of infinite sphere packings, Pacific J. Math. 50, 383–398.
Böröczky, K. (1971) Über die Newtonsche Zahl regulärer Vielecke, Periodica Math. Hungar. 1, 113–119.
Böröczky, K. (1974) Sphere packing in spaces of constant curvature I (Hungarian), Mat. Lapok 25, 265–306.
Böröczky, K. (1978) Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32, 243–261.
Böröczky, K. (1982) Closest packing and loosest covering of the space with balls, Studia Sci. Math. Hungar. 17 (to appear).
Böröczky, K. (1983) The problem of Tammes for n = 11, Studia Sci. Math. Hungar. (to appear).
Butler, G.J. (1972) Simultaneous packing and covering in Euclidean space, Proc. London Math. Soc. (3) 25, 721–735.
Cassels, J.W.S. (1959) An introduction to the geometry of numbers, Grundl. Math. Wiss. 99 Springer 1959, 1972. Berlin.
Chakerian, G.D. and Groemer, H. (1974) On classes of convex sets that permit plane coverings, Israel J. Math. 19, 305–311.
Chakerian, G.D. and Groemer, H. (1978) On coverings of euclidean space by convex sets, Pacific J. Math. 75, 77–80.
Chakerian, G.D. and Lange, L.H. (1971) Geometric extremum problems, Math. Mag. 44, 57–69.
Chvâtal, V. (1975) On a conjecture of Fejes T6th, Periodica Math. Hungar. 6, 357–362.
Cohn, M.J. (1976) Multiple lattice covering of space, Proc. London Math. Soc. III. 32, 117–132.
Conway, J.H., Parker, F.R.S. and Sloane, N.J.A. (1982) The covering radius of the Leech lattice, Proc. R. Soc. London A 380, 261–290.
Conway, J.H. and Sloane, N.J.A. (1982) Voronoi regions of lattices, 2nd moments of polytopes, and quantization, IEEE Transactions on Information Theory IT-28, 211–226.
Coxeter, H.S.M. (1948) Regular polytopes, Methuen, London.
Coxeter, H.S.M. (1954) Arrangements of equal spheres in non-Euclidean spaces, Acta Math. Acad. Sci. Hungar. 4, 263–274.
Coxeter, H.S.M. (1963) An upper bound for the number of equal nonoverlapping spheres that can touch another of the same size, Proceedings of Symposia in Pure Mathematics, Amer. Math. Soc. 7, Convexity, 53–71.
Coxeter, H.S.M., Few, L. and Rogers, C.A. (1959) Covering space by equal spheres, Mathematika 6, 147–157
Csóka, G. (1975a) On the number of congruent spheres hiding a given sphere (Russian), Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 18, 55–60.
Csóka, G. (1975b) On the densest packing of spheres in a cylinder in hyperbolic space (Russian), Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 18, 60–67.
Csóka, G. (1977) The number of congruent spheres that cover a given sphere of three-dimensional space, not less than 30 (Russian), Studia Sci. Math. Hungar. 12, 323–334.
Danzer, L. (1975) Endliche Punktmengen auf der S2 mit maximalem Mindestabstand. Kolloquium über Diskrete Geometrie, Salzburg 1975.
Davenport, H. and Hajós, G. (1951) Problem 35, Mat. Lapok 2, 68.
Davis, P.J. (1965) Simple quadratures in the complex plane, Pacific J. Math. 15, 813–824.
Delone, B.N., Dolbilin, N.P., Ryškov, S.S. and Stogrin, M.I. (1970) A new construction in the theory of lattice coverings of an n-dimensional space by equal spheres, Izv. Akad. Nauk SSSR Ser. Mat. 34, 289–298
Delone, B.N., Dolbilin, N.P., Ryškov, S.S. and Stogrin, M.I. (1970) A new construction in the theory of lattice coverings of an n-dimensional space by equal spheres, Math. USSR Izvestija 4, 293–302.
Delone, B.N. and Ryškov, S.S. (1963) Solution of the problem on the least dense lattice covering of a 4-dimensional space by equal spheres (Russian), Dokl. Akad. Nauk SSSR 152, 523–524
Delone, B.N. and Ryškov, S.S. (1963) Solution of the problem on the least dense lattice covering of a 4-dimensional space by equal spheres (Russian), Soviet Math. Dokl. 14, 1333–1334.
Delsarte, P., Goethals, J.M. and Seidel, J.J. (1977) Spherical codes and designs, Geometriae Dedicata 6, 363–388.
Dolbilin, N.P. (1974) On the covering radius of hyperbolically equivalent n-dimensional lattices and the boundedness of the index of a divided parallelepiped (Russian), Dokl. Akad. Nauk SSSR 214, 1002–1004
Dolbilin, N.P. (1974) On the covering radius of hyperbolically equivalent n-dimensional lattices and the boundedness of the index of a divided parallelepiped (Russian), Soviet Math. Dokl. 15, 277–279.
Dorninger, D. (1973) Überdeckung der Ebene durch inkongruente Kreise, Elern. Math. 28, 105–107.
Dowker, C.H. (1944) On minimum circumscribed polygons, Bull. Amer. Math. Soc. 50, 120–122.
Dumir, V.C. and Hans-Gill, R.J. (1972a) Lattice double coverings in the plane, Indian J. Pure Appl. Math. 3, 466–480.
Dumir, V.C. and Hans-Gill, R.J. (1972b) Lattice double packings in the plane, Indian J. Pure Appl. Math. 3, 481–487.
Dumir, V.C. and Khassa, D.S. (1973a) Saturated systems of symmetric convex domains; results of Eggleston, Bambah and Woods, Proc. Camb. Phil. Soc. 74, 107–116.
Dumir, V.C. and Khassa, D.S. (1973b) A conjecture of Fejes Tóth on saturated systems of circles, Proc. Camb. Phil. Soc. 74, 453–460.
Erdös, P. Few, L. and Rogers, C.A. (1964) The amount of overlapping in partial coverings of space by equal spheres, Mathematika 12, 171–184.
Erdös, P. and Pach, J. (1980) On a problem of L. Fejes Tóth, Discrete Math. 30, 103–109.
Erdös, P. and Rogers, C.A. (1962) Covering space with convex bodies, Acta Arith. 7, 281–285.
Fejes Tóth, G. (1971) Über Parkettierungen konstanter Nachbarnzahl, Studia Sci. Math. Hungar. 6, 133–135.
Fejes Tóth, G. (1972) Covering the plane by convex discs, Acta Math. Acad. Sci. Hungar. 23, 263–270.
Fejes Tóth, G. (1973) Sum of moments of convex polygons, Acta Math. Acad. Sci. Hungar. 24, 417–421.
Fejes Tóth, G. (1974) Solid sets of circles, Studia Sci. Math. Hungar. 9, 101–109.
Fejes Tóth, G. (1976a) Multiple packing and covering of the plane with circles, Acta Math. Acad. Sci. Hungar. 27, 135–140.
Fejes Tóth, G. (1976b) Research problem 18, Period. Math. Hungar. 7, 89–90.
Fejes Tóth, G. (1977a) On the intersection of a convex disc and a polygon, Acta Math. Acad. Sci. Hungar. 29, 149153.
Fejes Tóth, G. (1977b) A problem connected with multiple circle-packings and circle coverings, Studia Sci. Math. Hungar. 12, 447–456.
Fejes Tóth, G. (1978) Evading convex discs, Studia Sci. Math. Hungar. 13, 453–461.
Fejes Tóth, G. (1979) Multiple packing and covering of spheres, Acta Math. Acad. Sci. Hungar. 34, 165–176.
Fejes Tóth, G. (1980) On the section of a packing of equal balls, Studia Sci. Math. Hungar. 15.
Fejes Tóth, G. (1981) Ten-neighbour packing of equal balls, Period. Math. Hungar. 12, 125–127.
Fejes Tóth, G. (1982) Packing and covering with incongruent circles, Studia Sci. Math. Hungar. 17 (to appear).
Fejes Tóth, G. and Fejes Tóth, L. (1973) On totally separable domains, Acta Math. Acad. Sci. Hungar. 24, 229–232.
Fejes Tóth, G. and Fejes Tóth, L. (1980) Dictators on a planet, Studia Sci. Math. Hungar. 15, 313–316.
Fejes Tóth, G. and Fejes Tóth, L. (1982) Packing the plane with Minkowskian sums of convex sets, Studia Math. Hungar. 17 (to appear).
Fejes Tóth, G. and Florian, A. (1975) Mehrfache gitterförmige Kreis-and Kugelanordnungen, Monatsh. Math. 79, 13–20.
Fejes Tóth, L. (1953) On close-packings of spheres in spaces of constant curvature, Publ. Math. Debrecen 3, 158–167.
Fejes Tóth, L. (1966a) Mehrfache Kreisunterdeckungen and Kreisüberdeckungen auf der Kugel, Elem. Math. 21, 34–35.
Fejes Tóth, L. (1966b) On the permeability of a circle-layer, Studia Sci. Math. Hungar. 1, 5–10.
Fejes Tóth, L. (1967a) On the arrangement of houses in a housing estate, Studia Sci. Math. Hungar. 2, 37–42.
Fejes Tóth, L. (1967b) On the number of equal discs that can touch another of the same kind, Studia Sci. Math. Hungar. 2, 363–367.
Fejes Tóth, L. (1968a) On the permeability of a layer of parallelograms, Studia Sci. Math. Hungar. 3, 195–200.
Fejes Tóth, L. (1968b) Solid circle-packings and circle-coverings, Studia Sci. Math. Hungar. 3, 401–409.
Fejes Tóth, L. (1969a) Remarks on a theorem of R.M. R.binson, Studia Sci. Math. Hungar. 4, 441–445.
Fejes Tóth, L. (1969 b) Scheibenpackungen konstanter Nachbarnzahl, Acta Math. Acad. Sci. Hungar. 20, 375–381.
Fejes Tóth, L. (1970) Über eine affininvariante Masszahl bei Eipolyedern, Studia Sci. Math. Hungar. 5, 173–180.
Fejes Tóth, L. (1971a) Über Scheiben mit richtungsinvarianter Packungsdichte, Elem. Math. 26, 58–59.
Fejes Tóth, L. (1971b) The densest packing of lenses in the plane (Hungarian), Mat. Lapok 22, 209–213.
Fejes Tóth, L. (1972a) Lagerungen in der Ebene, auf der Kugel and im Raum, zweite Auflage, Grundl. Math. Wiss. 65, Springer-Verlag, Berlin-Heidelberg-New York.
Fejes Tóth, L. (1972b) A closing theorem (Hungarian), Mat. Lapok 23, 9–12.
Fejes Tóth, L. (1973a) On the density of a connected lattice of convex bodies, Acta Sci. Math. Hungar. 24, 373–376.
Fejes Tóth, L. (1973b) Five-neighbour packing of convex discs, Periodica Math. Hungar. 4, 221–229.
Fejes Tóth, L. (1975a) Research problem 13, Periodica Math. Hungar. 6, 197–199.
Fejes Tóth, L. (1975b) On Hadwiger numbers and Newton numbers of a convex body, Studia Sci. Math. Hungar. 10, 111–115.
Fejes Tóth, L. (1976) Close packing and loose covering with balls, Publ. Math. Debrecen 23, 323–326.
Fejes Tóth, L. (1977a) Dichteste Kugelpackung. Eine Idee von Gauß, Abh. Braunschweigischen Wiss. Gesell. 27, 311–321.
Fejes Tóth, L. (1977b) Illumination of convex discs, Acta Math. Acad. Sci. Hungar. 29, 355–360.
Fejes Tóth, L. (1977c) Research problem 21, Periodica Math. Hungar. 8, 103–104.
Fejes Tóth, L. (1978a) Remarks on the closest packing of convex discs, Comment. Math. Helvetici 33, 536–541.
Fejes Tóth, L. (1978b) Research problem 24, Periodica Math. Hungar. 9, 173–174.
Fejes Tóth, L. (1980a) Some density-bounds for packing and covering with incongruent circles, Studia Sci. Math. Hungar. 15, 63–70.
Fejes Tóth, L. (1980b) Approximation of convex domains by polygons, Studia Sci. Math. Hungar. 15, 133–138.
Fejes Tóth, L. (1980c) Packing and covering with convex discs, Studia Sci. Math. Hungar. 15, 93–100.
Fejes Tóth, L. (1980d) Solid packing of circles in the hyperbolic plane, Studia Sci. Math. Hungar. 15, 299–302.
Fejes Tóth, L. (1982a) Packing of r-convex discs, Studia. Sci. Math. Hungar. 17 (to appear).
Fejes Tóth, L. (1982b) Packing and covering with r-convex discs, Studia Sci. Math. Hungar. 17 (to appear).
Fejes Tóth, L. (1983a) Compact packing of circles, Studia Sci. Math. Hungar. (to appear).
Fejes Tóth, L. (1983b) Hiding a circle with circles ( Hungarian ), Mat. Lapok (to appear).
Fejes Tóth, L. (1983c) Density bounds for packing and covering with convex discs, Expositiones Mathematicae (to appear).
Fejes Tóth, L. (1983d) On the densest packing of convex discs, Mathematika (to appear).
Fejes Tóth, L. and Florian, A. (1982) Packing and covering with convex discs, Mathematika 29,.
Fejes Tóth, L. and Heppes, A. (1967) A variant of the problem of the thirteen spheres, Canad. J. Math. 19, 1092–1100
Fejes Tóth, L. and Heppes, A. (1977) A remark on the Hadwiger numbers of a convex disc, Studia Sci. Math. Hungar. 12, 409–412.
Fejes Tóth, L. and Heppes, A. (1980) Multi-saturated packings of circles, Studia Sci. Math. Hungar. 15, 303–307.
Fejes Tóth, L. and Makai, E. Jr. (1974) On the thinnest non-separable lattice of convex plates, Studia Sci. Math. Hungar. 9, 191–193.
Fejes Tóth, L. and Sauer, N. (1977) Thinnest packing of cubes with a given number of neighbours, Canad. Math. Bull. 20, 501–507.
Few, L. (1953) The double packing of spheres, J. London Math. Soc. 28, 297–304.
Few, L. (1964) Multiple packing of spheres, J. London Math. Soc. 39, 51–54.
Few, L. (1967a) Multiple packing of spheres: a survey, Proc Colloquium Convexity (Copenhagen 1965 ), K¢benhavns Univ. Mat. Inst., Kopenhagen, 88–93.
Few, L. (1967b) Double covering with spheres, Mathematika 14, 207–214.
Few, L. (1968) Double packing of spheres: A new upper bound, Mathematika 15, 88–92.
Few, L. and Kanagasabapathy, P. (1969) The double packing of spheres, J. London Math. Soc. 44, 141–146.
Florian, A. (1975a) Neuere Ergebnisse in der diskreten Geometrie, Berichte Math. Stat. Sekt. Forschungszentrum Graz, Bericht Nr. 45 (1975).
Florian, A. (1975b) Integrale auf konvexen Mosaiken, Periodica Math. Hungar. 6, 23–38.
Florian, A. (1975c) Reguläre hyperbolische Mosaike und Newtonshce Zahlen, Periodica Math. Hungar. 6, 97–101.
Florian, A. (1978a) Mehrfache Packung konvexer Körper, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 186, 373–384.
Florian, A. (1978b) On the permeability of layers of discs, Studia Sci. Math. Hungar. 13, 123–132.
Florian, A. (1979a) Neuere Entwicklung über reguläre Polyeder, Contributions to geometry, Proc. Symp. Siegen 1978, 238–247.
Florian, A. (1979b) Über die Durchlassigkeit gewisser Scheibenschichten, Österreich. Akad. Wiss. Math.-Natur. KI. Sitzungsber. II, 188, 417–427.
Florian, A. (1980a) Kugelwolken, Berichte Math.-Stat. Sekt. Forschungszentrum Graz, Bericht Nr. 143 (1980).
Florian, A. (1980b) Durchlässigkeit einer Scheibenschicht, Berichte Math.-Stat. Sekt. Forschungszentrum Graz, Bericht Nr. 144 (1980).
Florian, A. (1980c) Newtonsche und Hadwigersche Zahlen, Berichte Math. Stat. Sekt. Forschungszentrum Graz, Bericht Nr. 145 (1980).
Florian, A. (1980d) Über die Durchlässigkeit einer Schicht konvexer Scheiben, Studia Sci. Math. Hungar. 15, 201–213.
Florian, A. and Florian, H. (1975a) Reguläre hyperbolische Mosaike und Newtonsche Zahlen II, Periodica Math. Hungar. 6, 179–183.
Florian, A. and Florian, H. (1975b) Reguläre hyperbolische Mosaike und Newtonsche Zahlen III, Österreich. Akad. Wiss. Math.-Natur. KI. Sitzungber. II, 184, 29–40.
Florian, A., Hârs, L. and Molnar, J. (1979) On the p-systems of circles, Acta Math. Acad. Sci. Hungar. 34, 205–221.
Gâcs, P. (1972) Packing of convex sets in the plane with a great number of neighbours, Acta Math. Acad. Sci. Hungar. 23, 383–388.
Gauß, C.F. (1831) Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber, Göttingische gelehrte Anzeigen, 1831 Juli 9 = J. Reine Angew. Math. 20 (1840), 312–320
Gauß, C.F. (1831) Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber, Göttingische gelehrte Anzeigen, 1831 Juli 9 = J. Werke II, 188–196.
Gilbert, E.N. (1964) Randomly packed and solidly packed spheres, Canad. J. Math. 16, 286–298.
Goldberg, M. (1970) The packing of equal circles in a square, Math. Magazine, 43, 24–30.
Goldberg, M. (1971) Packing of 14, 16, 17 and 20 circles in a circle, Math. Magazine 44, 134–139.
Golser, G. (1977) Dichteste Kugelpackungen im Oktaeder, Studia Sci. Math. Hungar. 12, 337–343.
Groemer, H. (1960) Über die Einlagerung von Kreisen in einen konvexen Bereich, Math. Zeitschr. 73, 285–294.
Groemer, H. (1961) Abschätzungen für die Anzahl der konvexen Körper, die einen konvexen Körper berühren, Monatsh. Math. 65, 74–81.
Groemer, H. (1963) Existenzsätze für Lagerungen im Euklidischen Raum, Math. Zeitschr. 81, 260–278.
Groemer, H. (1966) Zusammenhängende Lagerungen konvexer Körper, Math. Zeitschr. 94, 66–78.
Groemer, H. (1968a) Existenzsätze für Lagerungen in metrischen Räumen, Monatsh. Math. 72, 325–334.
Groemer, H. (1968b) Einige Bemerkungen über zusammenhängende Lagerungen, Monatsh. Math. 72, 212–216.
Groemer, H. (1976a) On a covering property of convex sets, Proc. Amer. Math. Soc. 59, 346–352.
Groemer, H. (1976b) Some packing and covering problems, Amer. Math. Monthly 83, 726–727.
Groemer, H. (1979a) On multiple space subdivisions by zonotopes, Monatsh. Math. 86, 185–188.
Groemer, H. (1979b) Space coverings by translates of convex sets, Pacific J. Math. 82, 379–386.
Groemer, H. (1980) On finite classes of convex sets that permit space coverings, Amer. Math. Monthly 87, 286–288.
Groemer, H. (198la) On coverings of plane convex sets by translates of strips, Aequationes Math. 22, 215–222.
Groemer, H. (1981b) On coverings of convex sets by translates of slabs, Proc. Amer. Math. Soc. 82, 261–266.
Groemer, H. (1982) Covering and packing properties of bounded sequences of convex sets, Mathematika 29, 18–31.
Groemer, H. (1983a) On space coverings by unbounded convex sets, J. Combinatorial Theory, Ser. A.
Groemer, H. (1983b) On coverings of spheres by convex sets (to appear).
Groemer, H. and Heppes, A. (1975) Packing and covering properties of split disks, Studia Sci. Math. Hungar. 10, 185–189.
Gruber, P.M. (1979) Geometry of numbers, Contributions to geometry, Proc. Symp. Siegen 1978, 186–225.
Grünbaum, B. (1964) On a conjecture of Hadwiger, Pacific J. Math. 11, 215–219.
Grünbaum, B. and Shephard, G.C. (1983) Tilings and patterns.
Haas, A. (1976) Die dünnste siebenfache gitterförmige Überdeckung der Ebene durch kongruente Kreise, Dissertation, Wien.
Hadwiger, H. (1957) Über Treffanzahlen bei translationsgleichen Eikörpern, Archiv Math. 8, 212–213.
Hadwiger, H. (1970) Volumen und Oberfläche eines Eikörpers, der keine Gitterpunkte überdeckt, Math. Z. 116, 191–196.
Heppes, A. (1955) Über mehrfache Kreislagerungen, Elem. Math. 10, 125–127.
Heppes, A. (1959) Mehrfache gitterförmige Kreislagerungen in der Ebene, Acta Math. Acad. Sci. Hungar. 10, 141–148.
Heppes, A. (1960) Ein Satz über gitterförmige Kugelpackungen, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 3–4. 89–90.
Heppes, A. (1967) On the densest packing of circles not blocking each other, Studia Sci. Math. Hungar. 2, 257–263.
Hirst, K.E. (1967) The apollonian packing of circles, J. London Math. Soc. 42, 281–291.
Hlawka, E. (1944) Zur Geometrie der Zahlen, Math. Z. 49, 285–312.
Hollai, M. (1974) Lagerungen inkongruente Kugeln I, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 17, 73–90.
Hollai, M. (1975a) Doppelgitterförmige Lagerungen inkongruenter Kreise und Kugeln, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 18, 75–86.
Hollai, M. (1975b) Lagerungen inkongruenter Kugeln II, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 18, 101–116.
Hortobâgyi, I. (1971) Durchleutung gitterförmiger Kugelpackungen mit Lichtbündeln, Studia Sci. Math. Hungar. 6, 147–150.
Hortobâgyi, I. (1972) The Newton number of convex plane regions (Hungarian), Mat. Lapok 23, 313–317.
Hortobâgyi, I. (1975) Über die auf Scheibenklassen bezügliche Newtonsche Zahl der konvexen Scheiben, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 18, 123–127.
Hortobâgyi, I. (1976) Über die Durchlässigkeit einer aus Scheiben konstanter Breite bestehenden Schicht, Studia Sci. Math. Hung. 11, 383–387.
Hortobâgyi, I. (1977) Extreme problems connected with packings of sets of constant width, Thesis, Budapest (Hungarian).
Hortobâgyi, I. (1983a) Über die Anzahl derjenigen Richtungen in denen sich die gitterförmige Kugelpackungen durchleuchten lassen. Studia Sci. Math. Hungar. (to appear).
Hortobâgyi, I. (1983b) Durchleuchtung gitterförmiger Packungen von Körpern konstanter Breite mit Lichtbündeln, Studia Sci. Math. Hungar. (to appear).
Horvath, J. (1970) Über die Durchsichtigkeit gitterförmigen Kugelpackungen, Studia Sci. Math. Hungar. 5, 421–426.
Horvath, J. (1972a) On the densest packing of an n-dimensional cylinder with unit balls (Russian), Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 15, 139–143.
Horvath, J. (1972b) Remarks on the theory of clouds of spheres (Russian), Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 15, 145–150.
Horvath, J. (1974) Die Dichte einer Kugelpackung in einer 4-dimensionalen Schicht, Studia Sci. Math. Hungar. 5, 195–199.
Horvath, J. (1975) On the densest packing of unit balls in 3- and 4-dimensional slabs (Russian), Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 18, 171–176.
Horvath, J. (1977) Close lattice packings of balls in lattices of the first type (Russian), Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 20, 191–194.
Horvath, J. (1980) Close lattice packing of unit balls in the space E“ (Russian), Geometry of positive quadratic forms. Trudy Mat. Inst. Steklov. 152, 216–231.
Horvath, J. (1982) Über die Enge der gitterförmigen k-fachen Packungen, die Lockerheit der gitterförmigen k-fachen Überdeckungen und die k-Enge der gitterförmigen Punktmengen, Beiträge zur Algebra und Geometrie 16 (to appear)
Horvath, J. and Temesvâri, A.H. (1982) Über Dichte und Enge von doppelgitterförmigen zweifachen Kreispackungen, Studia Sci. Math. Hungar. 17 (to appear).
Hoylman, D.J. (1970) The densest lattice packing of tetrahedra, Bull. Amer. Math. Soc. 76, 135–137.
Ignat’ev, N.K. (1964) On a practical method of finding dense packings of n-spheres (Russian), Sibirsk. Mat. Z. 5, 815–819.
Ignat’ev, N.K. (1966) On finding dense packings of n-dimensional spheres (Russian), Sibirsk. Mat. Z. 7, 820–825.
Kabatjanskii, G.A. and Levenstein, V.I. (1978) Bounds for packings on the sphere and in space (Russian), Problemy Peredaci Informacii 14, 3–25
Kabatjanskii, G.A. and Levenstein, V.I. (1978) Bounds for packings on the sphere and in space (Russian), Problems of information transmission 14, 1–17.
Karabinta, A. and Székely, E. (1973) Sur les empilements optimaux de circles congruents sur la sphère, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 16, 143–154.
Kershner, R. (1939) The number of circles covering a set, Amer. J. Math. 61, 665–671.
Kleinschmidt, P., Pachner, U. and Wills, J.M. (1983) On L. Fejes Tóth’s sausage conjecture (to appear).
Kertész, G. (1982) On the problem of parasites (Hungarian), Dissertation, Budapest.
Korkine, A. and Zolotareff, G. (1872) Sur les formes quadratiques positives quaternaires, Math. Annalen 5, 581–583
Korkine, A. and Zolotareff, G. (1877) Sur les formes quadratiques, Math. Annalen 11, 242–249.
Kravitz, S. (1967) Packing cylinders into cylindrical containers, Math. Mag. 40, 65–71.
Krötenheerdt, O. (1974) Die flächenkleinsten Dreiecke, die zwei gegebene, sich von außen berührende Kreise enthalten, Beiträge zur Algebra und Geometrie 2, 145–154.
Krötenheerdt, O. (1975) Probleme der Lagerung von Figuren, Mitteilungen der Math. Gesselschaft der DDR 1, 65–74.
Krötenheerdt, O., Mammitzsch, C. and Richter, P. (1974) Die flächenkleinsten Vierecke, die zwei gegebene, sich von außen berührende Kreise enthalten, Beiträge zur Algebra und Geometrie 3, 167–183.
Kuperberg, W. (1982) Packings convex bodies in the plane with density greater than 3/4, Geometriae Dedicata 13, 149–155.
Lagrange, J.L. (1773) Recherches d’arithmetique, Nouv. Mem. Acad. Roy. Sc. Belle Lettres, Berlin 1773, 265–312
Lagrange, J.L. (1773) Recherches d’arithmetique, Oeuvres III, 693–758.
Lampert, D. and Csóka, G. (1973) On the density of regular system of circles (Russian), Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 16, 69–85.
Landau, H.J. and Slepain, O. (1966) On the optimality of the regular simplex code, Bell System Techn. J. 45, 1247–1272.
Larman, D.G. (1966a) On the exponent of the convergence of a packing of spheres, Mathematika 13, 57–59.
Larman, D.G. (1966b) A note on the Besicovich dimension of the closest packing of spheres in R, Proc. Camb. Phil. Soc. 62, 193–195.
Larman, D.G. (1967) On the Besicovitch dimension of the residual set of arbitrary packed discs in the plane, J. London Math. Soc. 42, 292–302.
Larman, D.G. (1968) On packings of unequal spheres in R,,, Canad. J. Math. 20, 967–969.
Leech, J. (1964) Some sphere packings in higher space, Canad. J. Math. 16, 657–682.
Leech, J. (1967a) Nones on sphere packings, Canad. J. Math. 19, 251–267.
Leech, J. (1967b) Five-dimensional nonlattice sphere packings, Canad. Math. Bull. 10, 387–393.
Leech, J. (1969) Six and seven dimensional nonlattice sphere packings, Canad. Math. Bull. 12, 151–155.
Leech, J. and Sloane, N.J.A. (1970a) New sphere packings in dimensions 9–15. Bull. Amer. Math. Soc. 76, 1006–1010.
Leech, J. and Sloane, N.J.A. (1970b) New sphere packings in more than thirty-two dimensions, Proc. Second Chapel Hill Conf. on Combin. Math. and Applic., Chapel Hill, N.C., 1970, 345–355.
Leech, J. and Sloane, N.J.A. (1971) Sphere packings and error-correcting codes, Canad. J. Math. 23, 718–745.
Lekkerkerker, C.G. (1969) Geometry of numbers, Bibl. Math. 8. Groningen: Wolters-Noordhoff et Amsterdam: North-Holland.
Levenštein, V.I. (1975) The maximal density of filling an n-dimensional Euclidean space with equal balls, Mat. Zametki 18, 301–311
Levenštein, V.I. (1975) The maximal density of filling an n-dimensional Euclidean space with equal balls, Math. Notes 18, 765–771.
Levenštein, V.I. (1979) Bounds for packing in n-dimensional Euclidean space (Russian), Dokl. Akad. Nauk SSSR 245, 1299–1303
Levenštein, V.I. (1979) Bounds for packing in n-dimensional Euclidean space (Russian), Soviet Math. Dokl. 20, 417–421.
Linhart, J. (1973a) Über einige Vermutungen von L. Fejes Tóth, Acta Math. Acad. Sci. Hungar. 24, 199–201.
Linhart, J. (1973b) Die Newtonsche Zahl von regelmäßigen Fünfecken, Periodica Math. Hungar. 4, 315–328.
Linhart, J. (1974) Endliche n-Nachbarnpackungen in der Ebene und auf der Kugel, Periodica Math. Hungar. 5, 301–306.
Linhart, J. (1975) Ein reguläres hyperbolisches Maximalmosaik, Österreich. Akad. Wiss. Math: Natur. K1. Sitzungsber. II, 184, 473–485.
Linhart, J. (1977a) Über die Kantenlängensumme von Dreieckspolyedern, Monatsh. Math. 83, 25–36.
Linhart, J. (1977b) Scheibenpackungen mit nach unten beschränkter Nachbarnzahl, Studia Sci. Math. Hungar. 12, 281–293.
Linhart, J. (1978) Closest packing and closest coverings by translates of a convex disc, Studia Sci. Math. Hungar. 13, 157–162.
Linhart, J. (1979) Kantenkrümmung und Umkugelradius konvexer Polyeder, Acta Math. Acad. Sci. Hungar. 34, 1–2.
Linhart, J. (1981) Die Beleuchtung von Kugeln, Geometriae Dedicata 10, 145–154.
Linhart, J. (1983) Eine Methode zur Berechnung der Dichte einer dichtesten gitterförmigen k-fachen Kreispackung, Arbeitsber. Math. Inst. Univ. Salzburg.
Lloyd, S.P. (1980) Hamming association schemes and codes on spheres, SIAM J. Math. Anal. 11, 488–505.
Makai, E. Jr. (1972) On centrosymmetric convex domains with a packing density independent of the direction, Studia Sci. Math. Hungar. 7, 423–424.
Makai, E. Jr. (1977) Packing problems in the Euclidean plane, Thesis, Budapest (Hungarian).
Makai, E. Jr. (1978) On the thinnest non-separable lattice of convex bodies, Studia Sci. Math. Hungar. 13, 19–27.
Makai, E. Jr. and Pach, J. (1983) Controlling of function classes and coverings of Euclidean space, Studia Sci. Math. Hungar. 18 (to appear).
Marley, G.C. (1973) Multiple subdivisions of E“, Rocky Mountain J. Math. 3, 583–589.
Marley, G.C. (1974) Multiple subdivisions of the plane, Math. Magazine 47, 202–206.
Matérn, B. and Person, O. (1965) On the extremum properties of the equilateral triangular lattice and the regular hexagonal network, Förtechning Over, Rapporter ack Uppsatser Stockholm, Nr. 7.
Mc Elice, P.J., Rodemick, E.R., Rumsey, H. Jr. and Welch, L.R. (1977) New upper bounds on the rate of a code via the Delsarte—MC Williams inequalities, IEEE Trans. Inform. Theory, 23, 157–166.
Melzak, Z.A. (1966) Infinite packings of disks, Canad. J. Math. 18, 838–852.
Melzak, Z.A. (1969) On the solid packing constant for circles, Math. Comp. 23, 169–172.
Mergelian, S.N. (1952) Uniform approximations to functions of a complex variable, Uspehi Mat. Nauk 7, 31–122.
Molnar, J. (1975) Sur les empilements optimeaux des sphères dans une sphère de l’espace à courbure constant à n-dimensions, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 18, 87–99.
Molnar, J. (1977) On the p-system of unit circles, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 20, 195–203.
Molnar, J. (1978) Packing of congruent spheres in a strip, Acta Math. Acad. Sci. Hungar. 31, 173–183.
Molnar, J. (1979) On the packing of unit circles in a convex domain, Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 22–23. 113–123.
Neville, E.H. (1915) On the solution of numerical functional equations, Proc. London Math. Soc. (2) 14, 308–326.
Odlyzko, A.M. and Sloane, N.J.A. (1979) New bounds on the number of unit spheres that can touch a unit sphere in n-dimensions, J. Combin. Theory Ser. A 26, 210–214.
Österreicher, F. and Linhart, J. (1981) Packungen kongruenter Stäbchen mit konstanter Nachbarnzahl, Elem. Math, 37, 5–16.
Pach, J. (1977) On the permeability problem, Studia Sci. Math. Hungar. 12, 419–424.
Pach, J. (1980) Decomposition of multiple packing and covering Diskrete Geometrie, 2. Kolloq., Inst. Math. Univ. Salzburg 1980, 169–178.
Pirl, U. (1967) Der Mindesabstand von n in Einheitskreisscheibe gelegenen Punkten, Math. Nachr. 40, 111–124.
Purdy, G.B. (1973) The lattice triple packing of spheres in Euclidean space, Trans. Amer. Math. Soc. 181, 457–470.
Rankin, R.A. (1947) On the closest packing of spheres in n-dimensions, Ann. Math. 48, 1062–1081.
Rankin, R.A. (1955) The closest packing of spherical caps in n-dimensions, Proc. Glasgow Math. Assoc. 2, 139–144.
Robinson, R.M. (1979) Multiple tilings of n-dimensional space by unit cubes, Math. Z. 166, 225–264.
Rogers, C.A. (1951) The closest packing of convex two-dimensional domains, Acta Math. 86, 309–321.
Rogers, C.A. (1957) A note on coverings, Mathematika 4, 1–6.
Rogers, C.A. (1958) The packing of equal spheres, Proc. London Math. Soc. (3) 8, 447–456.
Rogers, C.A. (1959) Lattice coverings of space, Mathematika 6, 33–39.
Rogers, C.A. (1963) Covering a sphere with spheres, Mathematika 10, 157–164.
Rogers, C.A. (1964) Packing and covering, Cambridge tracts 54, Cambridge Univ. Press, Cambridge.
Rogers, C.A. and Shephard, G.C. (1957) The difference body of a convex body, Arch. Math. 8, 220–233.
Ruda, M. (1969) Packing of squares in rectangles (Hungarian), MTA III. Oszt. Közleményei 19, 73–87.
Ryskov, S.S. (1974) Density of an (r,R)-system (Russian), Mat. Zametki 16, 474–454
Ryskov, S.S. (1974) Density of an (r,R)-system (Russian), Math. Notes 16, 855–858.
Ryskov, S.S. (1977) The geometry of positive quadratic forms (Russian), Proc. Intern. Congress of Math. (Vancouver 1974) I. 501–506
Ryskov, S.S. (1977) The geometry of positive quadratic forms (Russian), Amer. Math. Soc. Transi. II Ser. 109, 27–32.
Riskov, S.S. and Bavanouskiï, E.P. (1975) Solution of the problem of the least dense lattice covering of five-dimensional space by equal spheres (Russian), Dokl. Akad. Nauk SSSR 222, 39–42
Riskov, S.S. and Bavanouskiï, E.P. (1975) Solution of the problem of the least dense lattice covering of five-dimensional space by equal spheres (Russian), Soviet Math. Dokl. 6, 586–590.
Riskov, S.S. and Bavanouskiï, E.P. (1976) 29 c-types of n-dimensional primitive parallelohedra (with applications to the theory of coverings), Trudy Mat. Inst. Steklov 127 (Russian).
Ryškov, S.S. and Horvath, J. (1975) Estimation of the radius of a cylinder that can be imbedded in every lattice packing of n-dimensional unit balls (Russian), Mat. Zametki 17, 123–128
Ryškov, S.S. and Horvath, J. (1975) Estimation of the radius of a cylinder that can be imbedded in every lattice packing of n-dimensional unit balls (Russian), Math. Notes 17, 72–75.
Saaty, T.L. and Alexander, J.M. (1975) Optimization and the geometry of numbers: packing and covering, SIAM Review 17, 475–519.
Schaer, J. (1965) The densest packing of 9 circles in a square, Canad. Math. Bull. 8, 273–277.
Schaer, J. and Meir, A. (1965) On a geometric extremum problem, Canad. Math. Bull. 8, 21–27.
Schlüter, K. (1979) Kreispaçkungen in Quadraten, Elem. Math. 34, 12–14.
Schmidt, W.M. (1961) Zur Lagerung kongruenter Körper im Raum, Monatsh. Math. 65, 154–158.
Schmidt, W.M. (1963) On the Minkowski-Hlawka theorem, Illinois J. Math. 7, 18–23.
Schmitz, M. and Kirchner, K. (1982) Eine Verteilung von 13 Punkten auf einem Quadrat, Wiss. Zeitschrift Pädagogischen Hochschule Erfurt-Mühlhausen 18, 113–115.
Schopp, J. (1970) Über die Newtonsche Zahl einer Scheibe konstanter Breite, Studia Sci. Math. Hungar. 5, 475–478.
Shannon, C.E. (1959) Probability of error for optimal codes in a Gaussian channel, Bell System Tech. J. 38, 611–656.
Sidel’nikov, V.M. (1973) The densest packing of balls on the surface of the n-dimensional Euclidean sphere, and the number of vectors of a binary code with prescribed code distance (Russian), Dokl. Akad. Nauk SSSR 213, 1029–1032
Sidel’nikov, V.M. (1973) The densest packing of balls on the surface of the n-dimensional Euclidean sphere, and the number of vectors of a binary code with prescribed code distance (Russian), Soviet Math. Dokl. 14, 1851–1855.
Sidel’nikov, V.M. (1974) New estimates for the closest packing of spheres in n-dimensional Euclidean space (Russian), Mat. Sb. 95, 148–158.
Sidel’nikov, V.M. (1974) New estimates for the closest packing of spheres in n-dimensional Euclidean space (Russian), USSR Sb. 24, 147–157.
Škljarskii, D.O., Čencov, N.N. and Jaglom, I.M. (1974) Geometrical estimates and problems from combinatorial geometry (Russian), Library of the Mathematica Circle, No. 13, Izdat. “Nauka”, Moscow.
Sloane, N.J.A. (1972) Sphere packings constructed from BCH and Justesen codes, Mathematika 19, 183–190.
Sloane, N.J.A. (1977) Binary codes, lattices, and sphere packings, Combinatorial Surveys, P.J. Cameron Ed., Academic Press, London—New York, 117–164.
Sloane, N.J.A. (1978) Codes over GF(4) and complex lattices, J. Algebra 52, 168–181.
Sloane, N.J.A. (1979) Self-dual codes and lattices, Proceedings of Symposia in Pure Mathematics, Amer. Math. Soc. 34, Relation between éombinatorics and other parts of mathematics, 273–308.
Sloane, N.J.A. (1981) Tables of sphere packings and spherical codes, IEEE Trans. Inform. Theory, IT27, 327–338.
Sloane, N.J.A. (1982) Recent bounds for codes, sphere packings and related problems obtained by linear programming and other methods, Contemporary Mathematics 9, 153–185.
Smith, M.J. (1975) Packing translates of a compact set in Euclidean space, Bull. London Math. Soc. 7, 129–131.
Subak, H. (1960) Mehrfache gitterförmige Überdeckungen der Ebene durch Kreise, Dissetrtation, Wien.
Szegö, G. (1975) Orthogonal polynomials, Amer. Math. Soc. Colloquium Publications 29, Amer. Math. Soc. Providence, Rhode Island, Fourth edition.
Székely, E. (1974) Sur le problème de Tammes, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 17, 157–175.
Swinnerton-Dyer, H.P.F. (1953) Extremal lattices of convex bodies, Proc. Camb. Phil. Soc. 49, 161–162.
Tammela, P. (1970) An estimate of the critical determinant of a two-dimensional convex symmetric domain (Russian), Izv. VysII. Uöebn. Zaved. Mat. 12 (103), 103–107.
Tammes, R.M.L. (1930) On the origin of number and arrangement of the places of exit on the surface of pollen grains, Rec. Tray. Bot. Neerl. 27, 1–84.
Temesvâri, A.H. (1983a) Die dünnste gitterförmige 5-fache Kreisüberdeckung der Ebene, Studia Sci. Math. Hungar. 18 (to appear).
Temesvâri, A.H. (1983b) Die dünnste doppelgitterförmige 2-fache Kreisüberdeckung in der Ebene, Studia Sci. Math. Hungar. 18 (to appear).
Vâsârhelyi, E. (1983) Uber eine Überdeckung mit kongruenten Dreiecken, Beiträge zur Algebra und Geometrie (to appear).
Vermes, I. (1979) Ausfällungen der hyperbolischen Ebene durch kongruente Hyperzykelbereiche, Periodica Math. Hungar. 10, 217–229.
Vetöinkin, N.M. (1974) The packing of equal n-dimensional balls constructed from error-correcting codes (Russian), Studies in the geometry of positive quadratic forms, Ivanov Gos. Univ. Uéen. Zap. 89, 87–91.
Vetöinkin, N.M. (1980) Uniqueness of classes of positive quadratic forms, on which values of Hermite constants are reached for 6 < n < 8 (Russian), Geometry of positive quadratic forms. Trudy Mat. Inst. Steklov 152, 34–86.
Wegner, G. (1971) Bewegungsstabile Packungen konstanter Nachbarnzahl, Studia Sci. Math. Hungar. 6, 431–438.
Wegner, G. (1980) Zu einem ebenen Überdeckungsproblem, Studia Sci. Math. Hungar. 15.
Wegner, G. (1983) Über endliche Kreispackungen in der Ebene, Studia Sci. Math. Hungar. 18 (to appear).
Wengerodt, G. (1983a) Die dichteste Packung von sechzehn Kreisen in einem Quadrat, Beiträge zur Algebra and Geometrie (to appear).
Wengerodt, G. (1983b) Die dichteste Packung von 25 Kreisen in einem Quadrat, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (to appear).
Wesler, O. (1960) An infinite packing theorem for spheres, Proc. Amer. Math. Soc. 11, 324–326.
Wilker, J.B. (1967) Open disk packings of a disk, Canad. Math. Bull. 10, 395–415.
Wilker, J.B. (1973) The interval of disk packing exponents, Proc. Amer. Math. Soc. 41, 255–260.
Wilker, J.B. (1977) Sizing up a solid packing, Period. Math. Hungar. 8, 117–134.
Wills, J.M. (1968) Ein Satz über konvexe Mengen und Gitterpunkte, Monatsh. Math. 72, 451–463.
Wills, J.M. (1982) Research problem 30, Periodica Math. Hungar. 13, 75–76.
Wills, J.M. (1983) Research problem 31, Periodica, Math. Hungar.
Wyner, A.D. (1965) Capabilities of bounded discrepancy decoding, Bell System Tech. J. 44, 1061–1122.
Yang, L.J. (1980) Multiple lattice packings and coverings of spheres, Mh. Math. 89, 69–76.
Zamorzaev, A.M. (1965) On non-normal regular partitions of Euclidean space (Russian), Dokl. Akad. Nauk SSSR 161, 30–32.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer Basel AG
About this chapter
Cite this chapter
Tóth, G.F. (1983). New Results in the Theory of Packing and Covering. In: Gruber, P.M., Wills, J.M. (eds) Convexity and Its Applications. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5858-8_14
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5858-8_14
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5860-1
Online ISBN: 978-3-0348-5858-8
eBook Packages: Springer Book Archive