Abstract
A problem related to that of finding the Chebyshev center of a compact convex set in \(\mathbb R^n\) is considered, namely, the problem of calculating the center and the least positive ratio of a homothety under which the image of a given compact convex set in \(\mathbb R^n\) covers another given compact convex set. Both sets are defined by their support functions. A solution algorithm is proposed which consists in discretizing the support functions on a grid of unit vectors and reducing the problem to a linear programming problem. The error of the solution is estimated in terms of the distance between the given set and its approximation in the Hausdorff metric. For the stability of the approximate solution, it is essential that the sets be uniformly convex and a certain set in the dual space has a nonempty interior.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. L. Garkavi, “On the Chebyshev center and convex hull of a set,” Uspekhi Mat. Nauk 19 (6 (120)), 139–145 (1964).
A. R. Alimov and I. G. Tsar’kov, “Chebyshev centres, Jung constants, and their applications,” Russian Math. Surveys 74 (5), 775–849 (2019).
Alexey R. Alimov and Igor’ G. Tsar’kov, Geometric Approximation Theory (Springer, Cham, 2021).
A. Beck and A. C. Eldar, “Regularization in regression with bounded noise: a Chebyshev center approach,” SIAM J. Matrix Anal. Appl. 29 (2), 606–625 (2007).
Duzhi Wu, Jie Zhou, and Aiping Hu, “A new approximate algorithm for the Chebyshev center,” Automatica 49, 2483–2488 (2013).
V. Cerone, D. Piga, and D. Regruto, “Set-Membership Error-in-Variables Identification through Convex Relaxation Techniques,” IEEE Trans. Automat. Control 57 (2), 517–522 (2012).
Sheng Xu and R. M. Freund, “Solution Methodologies for the Smallest Enclosing Circle Problem,” Comput. Optim. Appl. 25, 283–292 (2003).
Y. Xia, M. Yang, and S. Wang, “Chebyshev center of the intersection of balls: complexity, relaxation and approximation,” Math. Program. Ser. A 187 (1-2), 287–315 (2021).
M. Milanese and R. Tempo, “Optimal algorithms theory for robust estimation and prediction,” IEEE Trans. Automat. Control 30 (8), 730–738 (1985).
N. D. Botkin and V. L. Turova-Botkina, “An algorithm for finding the Chebyshev center of a convex polyhedron,” Appl. Math. Optim. 29, 211–222 (1994).
S. I. Dudov and A. S. Dudova, “On the stability of inner and outer approximations of a convex compact set by a ball,” Comput. Math. Math. Phys. 47 (10), 1589–1602 (2007).
V. V. Abramova, S. I. Dudov, and M. A. Osiptsev, “The external estimate of the compact set by Lebesgue set of the convex function,” Izv. Saratov Univ. Math. Mech. Inform. 20 (2), 142–153 (2020).
S. I. Dudov, “Systematization of problems on ball estimates of a convex compactum,” Sb. Math. 206 (9), 1260–1280 (2015).
M. V. Balashov, “Approximate calculation of the Chebyshev center for a convex compact set in \(\mathbb R^n\),” J. Convex Anal. 29 (1), 157–164 (2022).
M. V. Balashov, “Chebyshev center and inscribed balls: properties and calculations,” Optim. Lett. 2021 (2021), https://doi.org/10.1007/s11590-021-01823-z.
Z. Xu, Y. Xia, and J. Wang, “Cheaper relaxation and better approximation for multi-ball constrained quadratic optimization and extension,” J. Global Opt. 80, 341–356 (2021).
L. Danzer, B. Grünbaum, and V. Klee, Helly’s Theorem and Its Relatives (Amer. Math. Soc., Providence, RI, 1963).
J. Diestel, Geometry of Banach spaces—Selected Topics (Springer- Verlag, Berlin, 1975).
E. S. Polovinkin and M. V. Balashov, Elements of Convex and Strongly Convex Analysis (Fizmatlit, Moscow, 2007) [in Russian].
M. V. Balashov and E. S. Polovinkin, “\(M\)-strongly convex subsets and their generating sets,” Sb. Math. 191 (1), 25–60 (2000).
E. S. Polovinkin, “Strongly convex analysis,” Sb. Math. 187 (2), 259–286 (1996).
M. V. Balashov, “On polyhedral approximations in an \(n\)-dimensional space,” Comput. Math. Math. Phys. 56 (10), 1679–1685 (2016).
M. V. Balashov and D. Repovš, “Polyhedral approximations of strictly convex compacta,” J. Math. Anal. Appl. 374, 529–537 (2011).
P. M. Gruber, “Approximation of convex bodies,” in Convexity and Its Applications (Birkhäuser, Basel, 1983), pp. 131–162.
D. Rosca, “New uniform grids on the sphere,” Astron. Astrophys. 520, A63 (2010).
D. Rosca and G. Plonka, “Uniform spherical grids via equal area projection from the cube to the sphere,” J. Comput. Appl. Math. 236 (3), 1033–1041 (2011).
Acknowledgments
The author is grateful to the referee for many valuable suggestions and comments, which helped to substantially improve the paper.
Funding
This work was supported by the Russian Science Foundation under grant 22-11-00042.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematicheskie Zametki, 2022, Vol. 112, pp. 337–349 https://doi.org/10.4213/mzm13537.
Rights and permissions
About this article
Cite this article
Balashov, M.V. Covering a Set by a Convex Compactum: Error Estimates and Computation. Math Notes 112, 349–359 (2022). https://doi.org/10.1134/S0001434622090024
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434622090024