View More View Less
  • 1 Hungarian Academy of Sciences Alfréd Rényi Institute of Mathematics 13-15 Reáltanoda u. 1053 Budapest Hungary
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

It is shown that in a packing of open circular discs with radii not exceeding 1, any two points lying outside the circles at distance d from one another can be connected by a path traveling outside the circles and having length at most

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{4} {\pi }d + O\left( {\sqrt d } \right)$$ \end{document}
. Given a packing of open balls with bounded radii in En and two points outside the balls at distance d from one another, the length of the shortest path connecting the two points and avoiding the balls is d + O(d/n) as d and n approaches infinity.

  • Baggett, D. R. and Bezdek, A., On a shortest path problem of G. Fejes Tóth, in: Discrete Geometry In Honor of W. Kuperberg’s 60th Birthday, Marcel Dekker, New York-Basel, 2003, 19–26.

    Bezdek A. , '', in Discrete Geometry In Honor of W. Kuperberg’s 60th Birthday , (2003 ) -.

  • Bezdek, A., On optimal route planning evading cubes in the three space, Beiträge Algebra Geom., 40 (1999), 79–87.

    Bezdek A. , 'On optimal route planning evading cubes in the three space ' (1999 ) 40 Beiträge Algebra Geom : 79 -87.

    • Search Google Scholar
  • Danzer, L., Personal communication.

  • Fejes Tóth, G., Evading convex discs, Studia Sci. Math. Hungar., 13 (1978), 453–461.

    Fejes Tóth G. , 'Evading convex discs ' (1978 ) 13 Studia Sci. Math. Hungar : 453 -461.

  • Fejes Tóth, L., Regular figures, A Pergamon Press Book The Macmillan Co., New York 1964.

    Fejes Tóth L. , '', in Regular figures , (1964 ) -.

  • Fejes Tóth, L., On the permeability of circle-layers, Studia Sci. Math. Hungar., 1 (1966), 5–10.

    Fejes Tóth L. , 'On the permeability of circle-layers ' (1966 ) 1 Studia Sci. Math. Hungar : 5 -10.

    • Search Google Scholar
  • Fejes Tóth, L., On the permeability of a layer of parallelograms, Studia Sci. Math. Hungar., 3 (1968), 5–10.

    Fejes Tóth L. , 'On the permeability of a layer of parallelograms ' (1968 ) 3 Studia Sci. Math. Hungar : 5 -10.

    • Search Google Scholar
  • Fejes Tóth, L., Research problem No. 24, Periodica Math. Hungar., 9 (1978), 173–174.

    Fejes Tóth L. , 'Research problem No. 24 ' (1978 ) 9 Periodica Math. Hungar : 173 -174.

  • Few, L., Multiple packing of spheres, J. London Math. Soc., 39 (1964), 51–54.

    Few L. , 'Multiple packing of spheres ' (1964 ) 39 J. London Math. Soc : 51 -54.

  • Hales, T., A proof of the Kepler conjecture, Annals of Math., 162 (2005), 065–1185.

    Hales T. , 'A proof of the Kepler conjecture ' (2005 ) 162 Annals of Math : 065 -1185.

  • Kabatjanskiĭ, G. A. and Levenšteĭn, V. I., Bounds on Packing on a sphere and in space (Russian), Problemy Peredaci Informacii, 14 (1978), 3–25, English translation in Problems Inform. Transmission, 14 (1978), 1–17.

    Levenšteĭn V. I. , 'Bounds on Packing on a sphere and in space (Russian) ' (1978 ) 14 Problemy Peredaci Informacii : 3 -25.

    • Search Google Scholar
  • Mitchell, J. S. B., Shortest paths and networks, in: Handbook of Discrete and Computational Geometry, Second edition, J. E. Goodman and J. O’Rourke eds., Chapma & Hall/ CRC, 607–641.

  • Pach, J., On the permeability problem, Studia Sci. Math. Hungar., 12 (1977), 419–424.

    Pach J. , 'On the permeability problem ' (1977 ) 12 Studia Sci. Math. Hungar : 419 -424.

  • Roldán-Pensado, E., Paths on the doubly covered region of a covering of the plane by discs, Studia Sci. Math. Hungar., to appear.