Bounded Contractibility of Strict Suns in Three-Dimensional Spaces

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BOUNDED CONTRACTIBILITY OF STRICT SUNS IN THREE-DIMENSIONAL SPACES A. R. Alimov

UDC 517.982.256+517.982.252

Abstract. A strict sun in a finite-dimensional (asymmetric) normed space X, dim X ≤ 3, is shown to be ˚ ˚ ˚ P -contractible, P -solar, B-infinitely connected, B-contractible, B-retract, and having a continuous additive (multiplicative) ε-selection for any ε > 0. A P -acyclic subset of a three-dimensional space is shown to have a continuous ε-selection for any ε > 0. For the dimension 3, the well-known Tsar’kov characterization of spaces, in which any bounded Chebyshev set is convex, is extended to the case of strict suns.

The best approximation, i.e., the distance from a given element x of a normed linear space X to a given nonempty set M ⊂ X is, by definition, ρ(x, M ) : = inf x − y. y∈M

The concepts and properties defined in terms of best approximation and, in particular, uniqueness and stability properties of elements of best approximation are called approximative. The first of such concepts is that of an element of best approximation, or nearest point. This is (for a given element x ∈ X) an element y ∈ M for which x − y = ρ(x, M ). The set of all nearest points (elements of best approximation) in M for a given x is denoted by PM x. In other words,   PM x : = y ∈ M | ρ(x, M ) = x − y . In the present paper, we are concerned with “solar” properties of subsets of normed linear spaces: properties of a mixed approximative-geometric characteristic. Throughout, X is a real normed linear space, and Xn is a Banach space X of finite dimension n. We also set: B(x, r) is the closed ball with center x and radius r; ˚ r) is the open ball with center x and radius r; B(x, S(x, r) is the sphere centered at x, of radius r. For brevity, the unit ball is denoted by B : = B(0, 1). We follow the terminology of the surveys [3, 4]. The main definitions are given below. Given ∅ = M ⊂ X, a point x ∈ X \ M is called a solar point if there exists a point y ∈ PM x = ∅ (called a luminosity point) such that   (1) y ∈ PM (1 − λ)y + λx for all λ ≥ 0 (this means that from y there is a “solar” ray through x such that y is a nearest point in M to any point from the ray). A point x ∈ X \ M is a strict protosolar point if PM x = ∅ and condition (1) holds for any point y ∈ PM x. A closed set M ⊂ X is a sun (respectively, a strict sun) if any point x ∈ X \ M is a solar point (respectively, a strict solar point) for M . A strict sun is a sun. A proximinal convex set is a strict sun. In a smooth space, any sun is convex. “Suns” have important characteristic features. They feature certain separation properties: a ball can be separated from such a set by means of a larger ball or a supporting cone. These properties are akin to the standard separability properties of convex sets by half-spaces (hyperplanes) and are closely related Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 1, pp. 3–11, 2018. c 2020 Springer Science+Business Media, LLC 1072–3374/20/2503–0385 

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to the classical Kolmogorov criterion (in so