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Monotone path-connectedness of R-weakly convex sets in the space C(Q)

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Abstract

A subset M of a normed linear space X is said to be R-weakly convex (R > 0 is fixed) if the intersection (D R (x, y) \ {x, y}) ∩ M is nonempty for all x, y ∈ M, 0 < ∥x − y∥ < 2R. Here D R (x, y) is the intersection of all the balls of radius R that contain x, y. The paper is concerned with connectedness of R-weakly convex sets in C(Q)-spaces. It will be shown that any R-weakly convex subset M of C(Q) is locally m-connected (locally Menger-connected) and each connected component of a boundedly compact R-weakly convex subset M of C(Q) is monotone path-connected and is a sun in C(Q). Also, we show that a boundedly compact subset M of C(Q) is R-weakly convex for some R > 0 if and only if M is a disjoint union of monotonically path-connected suns in C(Q), the Hausdorff distance between each pair of the components of M being at least 2R.

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    A. R. Alimov

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  1. A. R. Alimov
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Correspondence to A. R. Alimov.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 1, pp. 23–32, 2011/12.

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Alimov, A.R. Monotone path-connectedness of R-weakly convex sets in the space C(Q). J Math Sci 185, 360–366 (2012). https://doi.org/10.1007/s10958-012-0920-2

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  • DOI: https://doi.org/10.1007/s10958-012-0920-2

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