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 Tr. Mat. Inst. Steklova, 2018, Volume 303, Pages 246–257 (Mi tm3947)

Weakly monotone sets and continuous selection from a near-best approximation operator

I. G. Tsar'kov

Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia

Abstract: A new notion of weak monotonicity of sets is introduced, and it is shown that an approximatively compact and weakly monotone connected (weakly Menger-connected) set in a Banach space admits a continuous additive (multiplicative) $\varepsilon$-selection for any $\varepsilon >0$. Then a notion of weak monotone connectedness (weak Menger connectedness) of sets with respect to a set of $d$-defining functionals is introduced. For such sets, continuous $(d^{-1},\varepsilon )$-selections are constructed on arbitrary compact sets.

 Funding Agency Grant Number Russian Foundation for Basic Research 19-01-00332_a This work was supported by the Russian Foundation for Basic Research, project no. 19-01-00332-a.

DOI: https://doi.org/10.1134/S0371968518040180

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 303, 227–238

Bibliographic databases:

UDC: 517.982.256

Citation: I. G. Tsar'kov, “Weakly monotone sets and continuous selection from a near-best approximation operator”, Harmonic analysis, approximation theory, and number theory, Collected papers. Dedicated to Academician Sergei Vladimirovich Konyagin on the occasion of his 60th birthday, Tr. Mat. Inst. Steklova, 303, MAIK Nauka/Interperiodica, Moscow, 2018, 246–257; Proc. Steklov Inst. Math., 303 (2018), 227–238

Citation in format AMSBIB
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