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On local properties of spaces implying monotone path-connectedness of suns

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Abstract

The seminal A. L. Brown’s theorem asserts that in a finite-dimensional (BM)-space any sun is Menger connected, and in view of one earlier result of the author, is monotone path-connected. There is a well-known characterization of normed spaces of dimension 3 and 4 in which every Chebyshev set is convex, is that every exposed point of the unit sphere must be smooth. It turns out that, for a given set M, one can verify its convexity by testing for smoothness not all exposed points of the sphere, but only the so-called M-acting points of the sphere, i.e., the points such that an appropriate translation of a homothetic copy of the unit ball “touches” M with an “analogue” of this point. In the present paper, analogous to this result, we establish monotone path-connectedness of sets. Namely, we show that a sun M of a finite-dimensional normed space is monotone path-connected whenever any M-acting point of the unit sphere is a (BM)-point. We also give a characterization of three-dimensional polyhedral spaces in which every sun is monotone path-connected.

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Notes

  1. A set is m-connected (or Menger-connected) if, for any \(x,y\in M\), the set \(({\text {m}}(x,y)\!\setminus\! \{x,y\})\cap M\ne \varnothing\) (see [7, 12]).

  2. This property is satisfied, in particular, for the space \(c_0\) (see, for example, [7]).

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Acknowledgements

The author is grateful to the anonymous referee for finding some inaccuracies and misprints in the first version of this paper and for offering very useful remarks which allowed us to considerably improve the presentation of our results.

Funding

This research was carried out at Lomonosov Moscow State University with the financial support of the Russian Science Foundation (Grant no. 22-11-00129).

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia

    A. R. Alimov

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

    A. R. Alimov

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  1. A. R. Alimov
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Alimov, A.R. On local properties of spaces implying monotone path-connectedness of suns. J Anal 31, 2287–2295 (2023). https://doi.org/10.1007/s41478-023-00564-9

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