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Sphere Coverage in n Dimensions

Author

Listed:
  • Tatiana Tabirca

    (School of Computer Science and Information Technology, University College Cork, T12 XF62 Cork, Ireland)

  • Fangda Zou

    (School of Computer Science and Information Technology, University College Cork, T12 XF62 Cork, Ireland)

  • Sabin Tabirca

    (School of Computer Science and Information Technology, University College Cork, T12 XF62 Cork, Ireland
    Faculty of Mathematics and Informatics, Transilvania University of Brasov, 500036 Brasov, Romania)

Abstract

This paper presents some theoretical results on the sphere coverage problem in the n -dimensional space. These results refer to the minimal number of spheres, denoted by N k ( a ) , to cover a cuboid. The first properties outline some theoretical results for the numbers N k ( a ) , including sub-additivity and monotony on each variable. We use then these results to establish some lower and upper bounds for N k ( a ) , as well as for the minimal density of spheres to achieve k -coverage. Finally, a computation is proposed to approximate the N k ( a ) numbers, and some tables are produced to show them for 2D and 3D cuboids.

Suggested Citation

  • Tatiana Tabirca & Fangda Zou & Sabin Tabirca, 2024. "Sphere Coverage in n Dimensions," Mathematics, MDPI, vol. 12(23), pages 1-10, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:23:p:3772-:d:1533121
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    Keywords

    space coverage; minimal number of spheres;

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