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Continuous ε-selection and monotone path-connected sets

  • Volume 101, Number 6, June, 2017
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Abstract

The sets with continuous selection from near-best approximations and the monotone path-connected sets are studied; several examples of such sets are also considered.

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References

  1. A. R. Alimov and I. G. Tsar’kov, “Connectedness and other geometric properties of suns and Chebyshev sets,” Fundam. Prikl. Mat. 19 (4), 21–91 (2014) [J. Math. Sci. (New York) 217 (6), 683–730 (2016)].

    MathSciNet  MATH  Google Scholar 

  2. A. R. Alimov and I. G. Tsar’kov, “Connectedness and solarity in problems of best and near-best approximation,” UspekhiMat. Nauk 71 (1 (427)), 3–84 (2016) [RussianMath. Surveys 71 (1), 1–77 (2016)].

    Article  MathSciNet  MATH  Google Scholar 

  3. I. G. Tsar’kov, “Properties of sets that have a continuous selection from the operator Pδ,” Mat. Zametki 48 (4), 122–131 (1990) [Math. Notes 48 (3–4), 1052–1058 (1990)].

    MathSciNet  Google Scholar 

  4. S. V. Konyagin, “On continuous operators of generalized rational approximation,” Mat. Zametki 44 (3), 404 (1988).

    MathSciNet  MATH  Google Scholar 

  5. E. D. Livshits, “On the stability of the operator of ε-projection onto the set of splines in the space C[0, 1],” Izv. Ross. Akad. Nauk Ser. Mat. 67 (1), 99–130 (2003) [Izv. Math. 67 (1), 91–119 (2003)].

    Article  MathSciNet  Google Scholar 

  6. K. S. Ryutin, “Continuity of operators of generalized rational approximation in the space L1[0; 1],” Mat. Zametki 73 (1), 148–153 (2003) [Math. Notes 73 (1–2), 142–147 (2003)].

    Article  MathSciNet  Google Scholar 

  7. K. S. Ryutin, “Uniform continuity of generalized rational approximations,” Mat. Zametki 71 (2), 261–270 (2002) [Math. Notes 71 (1–2), 236–244 (2002)].

    Article  MathSciNet  MATH  Google Scholar 

  8. I. G. Tsar’kov, “Local and global continuous ε-selection,” Izv. Ross. Akad. Nauk Ser. Mat. 80 (2), 165–184 (2016) [Izv. Math. 80 (2), 442–461 (2016)].

    Article  MathSciNet  Google Scholar 

  9. I. G. Tsar’kov, “Continuous ε-selection,” Mat. Sb. 207 (2), 123–142 (2016) [Sb. Math. 207 (1–2), 267–285 (2016)].

    Article  MathSciNet  Google Scholar 

  10. I. G. Tsar’kov, “Properties of sets admitting a stable ε-selection,” Mat. Zametki 89 (4), 608–613 (2011) [Math. Notes 89 (3–4), 572–576 (2011)].

    Article  MathSciNet  Google Scholar 

  11. E. Michael, “Continuous selections. I,” Ann. of Math. (2) 63, 361–381 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  12. I. G. Tsar’kov, “Connectivity of certain classes of sets in Banach spaces,” Mat. Zametki 40 (2), 174–196 (1986).

    MathSciNet  Google Scholar 

  13. A. L. Brown, “Suns in normed linear spaces which are finite-dimensional,” Math. Ann. 279, 87–101 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. R. Alimov, “Monotone arcwise connectedness of Chebyshev sets in the space C(Q),” Mat. Sb. 197 (9), 3–18 (2006) [Sb. Math. 197 (9–10), 1259–1272 (2006)].

    Article  MathSciNet  Google Scholar 

  15. A. R. Alimov, “Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces,” Izv. Ross. Akad. Nauk Ser. Mat. 78 (4), 3–18 (2014) [Izv. Math. 78 (4), 641–655 (2014)].

    Article  MathSciNet  MATH  Google Scholar 

  16. A. R. Alimov and V. Yu. Protasov, “Separation of convex sets by extreme hyperplanes,” Fundam. Prikl. Mat. 17 (4), 3–12 (2012) [J. Math. Sci. (New York) 191 (5), 599–604 (2013)].

    MATH  Google Scholar 

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Correspondence to I. G. Tsar’kov.

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Original Russian Text © I. G. Tsar’kov, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 6, pp. 919–931.

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Tsar’kov, I.G. Continuous ε-selection and monotone path-connected sets. Math Notes 101, 1040–1049 (2017). https://doi.org/10.1134/S0001434617050315

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  • DOI: https://doi.org/10.1134/S0001434617050315

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