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Estimations of Covering Functionals of Convex Bodies Based on Relaxation Algorithm

Author

Listed:
  • Man Yu

    (School of Mathematics, North University of China, Taiyuan 030051, China)

  • Yafang Lv

    (School of Navigation, Wuhan University of Technology, Wuhan 430063, China)

  • Yanping Zhao

    (School of Mathematics, North University of China, Taiyuan 030051, China)

  • Chan He

    (School of Mathematics, North University of China, Taiyuan 030051, China)

  • Senlin Wu

    (School of Mathematics, North University of China, Taiyuan 030051, China)

Abstract

Estimating covering functionals of convex bodies is an important part of Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry. In this paper, we transform this problem into a vertex p -center problem (VPCP). An exact iterative algorithm is introduced to solve the VPCP by making adjustments to the relaxation-based algorithm mentioned by Chen and Chen in 2009. The accuracy of this algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean disc, simplices, and the regular octahedron. A better lower bound of the covering functional with respect to 7 of 3-simplices is presented.

Suggested Citation

  • Man Yu & Yafang Lv & Yanping Zhao & Chan He & Senlin Wu, 2023. "Estimations of Covering Functionals of Convex Bodies Based on Relaxation Algorithm," Mathematics, MDPI, vol. 11(9), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:2000-:d:1130925
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    References listed on IDEAS

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    1. R. Chen & G. Y. Handler, 1987. "Relaxation method for the solution of the minimax location‐allocation problem in euclidean space," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(6), pages 775-788, December.
    2. Chan He & Yafang Lv & Horst Martini & Senlin Wu, 2023. "A Branch-and-Bound Approach for Estimating Covering Functionals of Convex Bodies," Journal of Optimization Theory and Applications, Springer, vol. 196(3), pages 1036-1055, March.
    3. S. L. Hakimi, 1964. "Optimum Locations of Switching Centers and the Absolute Centers and Medians of a Graph," Operations Research, INFORMS, vol. 12(3), pages 450-459, June.
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