Abstract
Estimating covering functionals of convex bodies is not only interesting for its own, but also an important part of Chuanming Zong’s program to attack Hadwiger’s covering conjecture, a long-standing open problem from Discrete and Combinatorial Geometry. In general, it is difficult to determine exact values of the covering functionals of a convex body by theoretical analysis. Therefore we propose a global optimization algorithm based on the geometric branch-and-bound method. Numerical experiments have been carried out to estimate covering functionals of the Euclidean unit disc.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Berman, O., Drezner, Z., Krass, D.: Big segment small segment global optimization algorithm on networks. Networks 58(1), 1–11 (2011). https://doi.org/10.1002/net.20408
Bezdek, K.: Classical Topics in Discrete Geometry. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2010). https://doi.org/10.1007/978-1-4419-0600-7
Bezdek, K., Khan, M.: The geometry of homothetic covering and illumination. In: Conder M.D.E., Deza A., Weiss A.I. (eds.) Discrete Geometry and Symmetry, Springer Proceedings of Mathematical Statistics, vol. 234, pp. 1–30. Springer, Cham (2018)
Boltyanski, V., Martini, H., Soltan, P.: Excursions into Combinatorial Geometry. Springer-Verlag, Berlin (1997). https://doi.org/10.1007/978-3-642-59237-9
Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)
Drezner, Z., Suzuki, A.: The big triangle small triangle method for the solution of non-convex facility location problems. Oper. Res. 52(1), 128–135 (2004). https://doi.org/10.1287/opre.1030.0077
Hansen, P., Peeters, D., Thisse, J.: On the location of an obnoxious facility. Sistemi Urbani 3, 299–317 (1981)
He, C., Martini, H., Wu, S.: On covering functionals of convex bodies. J. Math. Anal. Appl. 437(2), 1236–1256 (2016). https://doi.org/10.1016/j.jmaa.2016.01.055
Henk, M.: Löwner-John ellipsoids. Doc. Math. (Extra vol.: Optimization stories), pp. 95–106 (2012)
Martini, H., Soltan, V.: Combinatorial problems on the illumination of convex bodies. Aequ. Math. 57(2–3), 121–152 (1999). https://doi.org/10.1007/s000100050074
Martini, H., Swanepoel, K., Weiß, G.: The geometry of Minkowski spaces–a survey. I. Expo. Math. 19(2), 97–142 (2001). https://doi.org/10.1016/S0723-0869(01)80025-6
Plastria, F.: GBSSS, the generalized big square small square method for planar single facility location. Eur. J. Oper. Res. 62(2), 163–174 (1992). https://doi.org/10.1016/0377-2217(92)90244-4
Schöbel, A., Scholz, D.: The big cube small cube solution method for multidimensional facility location problems. Comput. Oper. Res. 37(1), 115–122 (2010). https://doi.org/10.1016/j.cor.2009.03.031
Scholz, D.: Deterministic Global Optimization, Springer Optimization and Its Applications. Springer, New York (2012). https://doi.org/10.1007/978-1-4614-1951-8
Soltan, V.: Affine diameters of convex bodies-a survey. Expo. Math. 23(1), 47–63 (2005). https://doi.org/10.1016/j.exmath.2005.01.019
Wu, S., Fan, B., He, C.: Covering functionals of Minkowski sums and direct sums of convex bodies. Math. Inequal. Appl. 23(3), 1145–1154 (2020). https://doi.org/10.7153/mia-2020-23-88
Wu, S., Fan, B., He, C.: Covering a convex body vs. covering the set of its extreme points. Beitr. Algebr. Geom. 62(1), 281–290 (2021). https://doi.org/10.1007/s13366-020-00516-5
Wu, S., He, C.: Covering functionals of convex polytopes. Linear Algebr. Appl. 577, 53–68 (2019). https://doi.org/10.1016/j.laa.2019.04.022
Wu, S., Xu, K.: Covering functionals of cones and double cones. J. Inequal. Appl. 186, 9 (2018). https://doi.org/10.1186/s13660-018-1785-9
Zahn, C.: Black box maximization of circular coverage. J. Res. Nat. Bur. Stand. B 66B(4), 181–216 (1962)
Zong, C.: A quantitative program for Hadwiger’s covering conjecture. Sci. China Math. 53(9), 2551–2560 (2010). https://doi.org/10.1007/s11425-010-4087-3
Acknowledgements
The authors are supported by the National Natural Science Foundation of China (Grant Nos. 12071444 and 12201581), the Fundamental Research Program of Shanxi Province of China (Grant Nos. 201901D111141 and 202103021223191).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by Anita Schöbel.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
He, C., Lv, Y., Martini, H. et al. A Branch-and-Bound Approach for Estimating Covering Functionals of Convex Bodies. J Optim Theory Appl 196, 1036–1055 (2023). https://doi.org/10.1007/s10957-022-02146-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-022-02146-4