Abstract

The paper is concerned with new concepts of the max-approximation theory related to solar properties of sets: a max-sun, a local max-sun, max-Chebyshev set, and a max-attractor. Some results on max-solarity are established for uniquely remotal sets (such sets are called absolute max-Chebyshev sets). An example of a nonsingleton (nonclosed) uniquely remotal set on an asymmetrically normed plane is constructed, which gives a negative answer in the problem of whether a uniquely remotal set is a singleton (the unique farthest point problem). Results are obtained both in symmetrically and asymmetrically normed spaces.

Author information

Contact details are reproduced from the original publication and may be historical.

Igor' G. Tsar'kov

Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia

igtsarkov@yandex.ru

Keywords

  • Max-approximation
  • farthest point
  • absolutely max-Chebyshev set
  • uniquely remotal set
  • max-sun
  • local max-sun
  • point of max-luminosity
  • Chebyshev center

Mathematics Subject Classification

41A65, 52A21, 41A28

References

  1. A. R. Alimov: Solarity of sets in max-approximation problems, J. Fixed Point Theory Appl. 21/3 (2019), art. no. 76.https://doi.org/10.1007/s11784-019-0713-zGoogle Scholar
  2. A. R. Alimov: Solarity of Chebyshev sets in dual spaces and uniquely remotal sets, Lobachevskii J. Math. 42/4 (2021) 785–790.https://doi.org/10.1134/S199508022104003XGoogle Scholar
  3. A. R. Alimov, I. G. Tsar’kov: Connectedness and solarity in problems of best and near-best approximation, Russian Math. Surveys 71/1 (2016) 1–77.https://doi.org/10.1070/RM9698Google Scholar
  4. A. R. Alimov, I. G. Tsar’kov: Chebyshev centres, Jung constants, and their applications, Russian Math. Surveys 74/5 (2019) 775–849.https://doi.org/10.1070/RM9839Google Scholar
  5. A. R. Alimov, I. G. Tsar’kov: Geometric Approximation Theory, Springer, Cham (2021).https://doi.org/10.1007/978-3-030-90951-2Google Scholar
  6. A. R. Alimov, I. G. Tsar’kov: Approximatively compact sets in asymmetric Efimov-Stechkin spaces and convexity of almost suns, Math. Notes 110/6 (2021) 947–951.https://doi.org/10.1134/S0001434621110316Google Scholar
  7. A. R. Alimov, I. G. Tsar’kov: Ball-complete sets and solar properties of sets in asymmetric spaces, Results Math. 77/2 (2022), art. no. 86.https://doi.org/10.1007/s00025-022-01619-2Google Scholar
  8. M. V. Balashov: Approximate calculation of the Chebyshev center for a convex compact set in Rn, J. Convex Analysis 29/1 (2022) 157–164.https://doi.org/10.68381/jca29009Google Scholar
  9. M. V. Balashov, G. E. Ivanov: The farthest and the nearest points of sets, J. Convex Analysis 25/3 (2018) 1019–103.https://doi.org/10.68381/jca25065Google Scholar
  10. M. Baronti, P. L. Papini: Remotal sets revisited, Taiwan. J. Math. 5 (2001) 367–373.https://doi.org/10.11650/twjm/1500407343Google Scholar
  11. T. Bendit: Metric Projections in Inner Product Spaces, PhD Thesis, University of Newcastle, Australia (2021).Google Scholar
  12. Ş. Cobzaş: Functional Analysis in Asymmetric Normed Spaces, Birkhäuser/Springer, Basel (2013).https://doi.org/10.1007/978-3-0348-0478-3Google Scholar
  13. V. Donjuán, N. Jonard-Pérez: Separation axioms and covering dimension of asymmetric normed spaces, Quaest. Math. 43/4 (2020) 467–491.https://doi.org/10.2989/16073606.2019.1581298Google Scholar
  14. J. Dugundji: Topology, Allyn and Bacon, New York (1966).Google Scholar
  15. R. Huotari, M. P. Prophet, J. Shi: A farthest-point characterisation of the relative Chebyshev centre, Bull. Aust. Math. Soc. 54/1 (1996) 27–33.https://doi.org/10.1017/S0004972700015045Google Scholar
  16. G. E. Ivanov: Farthest points and strong convexity of sets, Math. Notes 87/3 (2010) 355–366.https://doi.org/10.1134/S0001434610030065Google Scholar
  17. B. G. Merino, T. Jahn, C. Richter: Uniqueness of circumcenters in generalized Minkowski spaces, J. Approx. Theory 237 (2019) 153–159.https://doi.org/10.1016/j.jat.2018.09.005Google Scholar
  18. W. B. Moors: Nearly Chebyshev sets are almost convex, Set-Valued Var. Analysis 26/1 (2018) 67–76.https://doi.org/10.1007/s11228-017-0445-4Google Scholar
  19. T. S. Motzkin, E. G. Straus, F. A. Valentine: The number of farthest points, Pacific J. Math. 3 (1953) 221–232.https://doi.org/10.2140/pjm.1953.3.221Google Scholar
  20. T. D. Narang, Sangeeta: On singletonness of remotal and uniquely remotal sets, Bull. Belgian Math. Soc. Simon Stevin 18 (2011) 113–120.https://doi.org/10.36045/bbms/1299766492Google Scholar
  21. B. B. Panda: On farthest point property, Indian J. Pure Appl. Math. 19/3 (1988) 277–283.Google Scholar
  22. S. Park: Applications of fixed point theorems for acyclic maps – a review, Vietnam J. Math. 37 (2009) 419–441.Google Scholar
  23. I. G. Tsar’kov: Metric max-distance function in max-approximation problems, Russ. J. Math. Phys. 26 (2019) 219–226.https://doi.org/10.1134/S1061920819020092Google Scholar
  24. I. G. Tsar’kov: Uniform convexity in nonsymmetric spaces, Math. Notes 110/5 (2021) 773–783.https://doi.org/10.1134/S0001434621110146Google Scholar

Suggested citation

A. R. Alimov, I. G. Tsar'kov. “Max-Solar Properties of Sets in Normed and Asymmetrically Normed Spaces.” Journal of Convex Analysis 30 (2023), No. 1, 159–174. https://doi.org/10.68381/jca30010

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Published by Heldermann Verlag, 2023. Rights now held by Banach Press.