
This article is cited in 3 scientific papers (total in 3 papers)
Convexity and monotone linear connectivity of sets with a continuous metric projection in threedimensional spaces
A. R. Alimov^{abc} ^{a} Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
^{b} Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
^{c} Moscow Center for Fundamental and Applied Mathematics
Abstract:
A continuous curve $k( {\cdot} )$ in a normed linear space $X$ is called monotone if the function $f(k(\tau))$ is monotone with respect to $\tau$ for any extreme functional $f$ of the unit dual sphere $S^*$. A closed set is monotone pathconnected if any two points from it can be connected by a continuous monotone curve lying in this set. We prove that in a threedimensional Banach space any closed set with lower semicontinuous metric projection is monotone pathconnected if and only if the norm of the space is either cylindrical or smooth. This result partially extends a recent result of the author of this paper and B. B. Bednov, who characterized the threedimensional spaces in which any Chebyshev set is monotone pathconnected. We show that in a finitedimensional Banach space any closed set with lower semicontinuous (continuous) metric projection is convex if and only if the space is smooth. A number of new properties of strict suns in threedimensional spaces with cylindrical norm is put forward. It is shown that in a threedimensional space with cylindrical norm a closed set $M$ with lower semicontinuous metric projection is a strict sun. Moreover, such a set $M$ has contractible intersections with closed balls and possesses a continuous selection of the metric projection operator. Our analysis depends substantially on the novel machinery of approximation of the unit sphere by polytopes built from tangent directions to the unit sphere.
Keywords:
set with continuous metric projection, Chebyshev set, sun, monotone pathconnected set.
DOI:
https://doi.org/10.21538/0134488920202622846
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UDC:
517.977
MSC: 41A65 Received: 19.12.2019 Revised: 28.01.2020 Accepted:10.02.2020
Citation:
A. R. Alimov, “Convexity and monotone linear connectivity of sets with a continuous metric projection in threedimensional spaces”, Trudy Inst. Mat. i Mekh. UrO RAN, 26, no. 2, 2020, 28–46
Citation in format AMSBIB
\Bibitem{Ali20}
\by A.~R.~Alimov
\paper Convexity and monotone linear connectivity of sets with a continuous metric projection in threedimensional spaces
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2020
\vol 26
\issue 2
\pages 2846
\mathnet{http://mi.mathnet.ru/timm1719}
\crossref{https://doi.org/10.21538/0134488920202622846}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=4131090}
\elib{https://elibrary.ru/item.asp?id=42950645}
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This publication is cited in the following articles:

A. R. Alimov, “Characterization of Sets with Continuous Metric Projection in the Space $\ell^\infty_n$”, Math. Notes, 108:3 (2020), 309–317

A. R. Alimov, “Geometric construction of Chebyshev sets and suns in threedimensional spaces with cylindrical norm”, Moscow University Mathematics Bulletin, Moscow University Mеchanics Bulletin, 75:5 (2020), 209–215

A. R. Alimov, B. B. Bednov, “Monotone pathconnectedness of Chebyshev sets in threedimensional spaces”, Sb. Math., 212:5 (2021), 636–654

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