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Chebyshev Sets with Piecewise Continuous Metric Projection

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Abstract

We study the properties of Chebyshev sets composed of at most countably many sets with a continuous metric projection. Local solarity is established in neighborhoods on which the metric projection is single-valued and continuous in uniformly convex spaces. As examples of applications of the results obtained, we consider generalized fractions and products, as well as ridge functions.

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References

  1. I. G. Tsar’kov, “Properties of suns in the spaces \(L^1\) and \(C(Q)\),” Russ. J. Math. Phys. 28 (3), 398–405 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. R. Alimov and I. G. Tsar’kov, “Connectedness and solarity in problems of best and near-best approximation,” Russ. Math. Surv. 71 (1), 1–77 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  3. V. S. Balaganskij and L. P. Vlasov, “The problem of convexity of Chebyshev sets,” Russ. Math. Surv. 51 (6), 1127–1190 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. R. Alimov and I. G. Tsar’kov, “Connectedness and other geometric properties of suns and Chebyshev sets,” J. Math. Sci. (New York) 217 (4), 683–730 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. R. Alimov and I. G. Tsar’kov, “Ball-complete sets and solar properties of sets in asymmetric spaces,” Results Math. 77 (2), 86 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. R. Alimov and I. G. Tsar’kov, Geometric Approximation Theory (Springer- Verlag, Cham, 2021).

    Book  MATH  Google Scholar 

  7. A. R. Alimov and I. G. Tsar’kov, “Suns, moons, and \(\mathring B\)-complete sets in asymmetric spaces,” Set-Valued Var. Anal. 30 (3), 1233–1245 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. R. Alimov and I. G. Tsar’kov, “\(B\)-Complete sets: Approximative and structural properties,” Sib. Math. J. 63, 412–420 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  9. I. G. Tsar’kov, “Continuity of the metric projection, structural and approximate properties of sets,” Math. Notes 47 (2), 218–227 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  10. I. G. Tsar’kov, “The distance function and boundedness of diameters of the nearest elements,” in Modern Methods in Operator Theory and Harmonic Analysis, Springer Proc. Math. Stat. (Springer- Verlag, Cham, 2019), Vol. 291, pp. 263–272.

    Chapter  Google Scholar 

  11. I. G. Tsar’kov, “Approximative compactness and nonuniqueness in variational problems, and applications to differential equations,” Sb. Math. 202 (6), 909–934 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  12. I. G. Tsar’kov, “Nonunique solvability of certain differential equations and their connection with geometric approximation theory,” Math. Notes 75 (2), 259–271 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  13. I. G. Tsar’kov, “Singular sets of surfaces,” Russ. J. Math. Phys. 24 (2), 263–271 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  14. I. G. Tsar’kov, “Geometry of the singular set of hypersurfaces and the eikonal equation,” Russ. J. Math. Phys. 29 (2), 240–248 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  15. N. V. Efimov and S. B. Stechkin, “Approximate compactness and Chebyshev sets,” Sov. Math. Dokl. 2, 1226–1228 (1961).

    MATH  Google Scholar 

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Funding

The study was carried out at Lomonosov Moscow State University and was supported by the Russian Science Foundation under grant no. 22-21-00204, https://rscf.ru/en/project/22-21-00204/.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    I. G. Tsar’kov

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    I. G. Tsar’kov

Authors
  1. I. G. Tsar’kov
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Correspondence to I. G. Tsar’kov.

Additional information

Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 905–917 https://doi.org/10.4213/mzm13628.

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Tsar’kov, I.G. Chebyshev Sets with Piecewise Continuous Metric Projection. Math Notes 113, 840–849 (2023). https://doi.org/10.1134/S0001434623050255

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  • DOI: https://doi.org/10.1134/S0001434623050255

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