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Izv. RAN. Ser. Mat., 2014, Volume 78, Issue 4, Pages 3–18 (Mi izv8128)  

This article is cited in 11 scientific papers (total in 11 papers)

Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces

A. R. Alimov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We prove that every boundedly compact $\operatorname{m}$-connected (Menger-connected) set is monotone path-connected and is a sun in a broad class of Banach spaces (in particular, in separable spaces). We show that the intersection of a boundedly compact monotone path-connected ($\operatorname{m}$-connected) set with a closed ball is cell-like (of trivial shape) and, in particular, acyclic (contractible in the finite-dimensional case) and is a sun. We also prove that every boundedly weakly compact $\operatorname{m}$-connected set is monotone path-connected. In passing, we extend the Rainwater–Simons weak convergence theorem to the case of convergence with respect to the associated norm (in the sense of Brown).

Keywords: sun, acyclic set, cell-like set, monotone path-connected set, Menger connectedness, $d$-convexity, Menger convexity, Rainwater–Simons theorem.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00022


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English version:
Izvestiya: Mathematics, 2014, 78:4, 641–655

Bibliographic databases:

Document Type: Article
MSC: Primary 41A65; Secondary 52A01
Received: 15.05.2013
Revised: 18.10.2013

Citation: A. R. Alimov, “Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces”, Izv. RAN. Ser. Mat., 78:4 (2014), 3–18; Izv. Math., 78:4 (2014), 641–655

Citation in format AMSBIB
\by A.~R.~Alimov
\paper Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces
\jour Izv. RAN. Ser. Mat.
\yr 2014
\vol 78
\issue 4
\pages 3--18
\jour Izv. Math.
\yr 2014
\vol 78
\issue 4
\pages 641--655

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    This publication is cited in the following articles:
    1. A. R. Alimov, “The Rainwater–Simons weak convergence theorem for the Brown associated norm”, Eurasian Math. J., 5:2 (2014), 126–131  mathnet
    2. A. R. Alimov, “Vypuklost ogranichennykh chebyshevskikh mnozhestv v konechnomernykh prostranstvakh s nesimmetrichnoi normoi”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:4(2) (2014), 489–497  mathnet
    3. A. R. Alimov, I. G. Tsar'kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, J. Math. Sci., 217:6 (2016), 683–730  mathnet  crossref  mathscinet
    4. A. R. Alimov, “On finite-dimensional Banach spaces in which suns are connected”, Eurasian Math. J., 6:4 (2015), 7–18  mathnet
    5. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. I. G. Tsar'kov, “Local and global continuous $\varepsilon$-selection”, Izv. Math., 80:2 (2016), 442–461  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. I. G. Tsar'kov, “Continuous $\varepsilon$-Selection and Monotone Path-Connected Sets”, Math. Notes, 101:6 (2017), 1040–1049  mathnet  crossref  crossref  mathscinet  isi  elib
    8. A. R. Alimov, “Selections of the metric projection operator and strict solarity of sets with continuous metric projection”, Sb. Math., 208:7 (2017), 915–928  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. I. G. Tsar'kov, “New Criteria for the Existence of a Continuous $\varepsilon$-Selection”, Math. Notes, 104:5 (2018), 727–734  mathnet  crossref  crossref  isi  elib
    10. A. R. Alimov, “Selections of the best and near-best approximation operators and solarity”, Proc. Steklov Inst. Math., 303 (2018), 10–17  mathnet  crossref  crossref  isi  elib
    11. I. G. Tsar'kov, “Weakly monotone sets and continuous selection from a near-best approximation operator”, Proc. Steklov Inst. Math., 303 (2018), 227–238  mathnet  crossref  crossref  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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