The connectivity and approximative properties of sets in linear normed spaces

Koshcheev, V. A.

doi:10.1007/bf01161866

Cited by 18 publications

(4 citation statements)

References 3 publications

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“…More details about Chebyshev sets, suns, and strict suns can be found in [1][2][3][4][5][6][7][8][9][10][11]. If x ∈ X and r > 0, then by B(x, r),B(x, r), and S(x, r) we denote the closed ball, the open ball, and the sphere with center x and radius r, respectively; to simplify the notation, we write B = B(0, 1) andB =B(0, 1).…”

Section: Definitions and Auxiliary Resultsmentioning

confidence: 99%

The geometric structure of Chebyshev sets in ℓ∞(n)

Алимов

2005

Funct Anal Its Appl

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A subset M of a normed linear space X is called a Chebyshev set if each x ∈ X has a unique nearest point in M . We characterize Chebyshev sets in ∞ (n) in geometric terms and study the approximative properties of sections of Chebyshev sets, suns, and strict suns in ∞ (n) by coordinate subspaces.points nearest to x in M consists of one point. For example, the subspace of polynomials of degree m in C[0, 1] and a line on a Euclidean plane are Chebyshev sets.There are two main results in this paper. Theorem 1 describes the approximative properties of intersections of Chebyshev sets, suns, and strict suns in ∞ (n) with coordinate subspaces. Theorem 2 characterizes Chebyshev sets in ∞ (n) in geometric terms.The paper has the following structure. In Sec. 1, necessary definitions and auxiliary results are given. Theorems 1 and 2 are stated in Secs. 2 and 3, respectively. Section 4 contains the proofs of Theorems 1 and 2. Definitions and Auxiliary ResultsLet us introduce definitions needed in the statement of the main results. Let k ∈ N and 1 k n. By cAff k (R n ) we denote the set of k-dimensional affine subspaces of R n parallel to k-dimensional faces * * of the unit ball B in ∞ (n). In other words, cAff k (R n ) consists of affine subspaces of the form lin{e i 1 , . . . , e i k : 1 i 1 < · · · < i k n} + x, x ∈ R n , where e 1 , . . . , e n is the standard basis in R n . The elements of cAff k (R n ) will be called affine coordinate subspaces. (Sometimes the word "affine" will be omitted for brevity.) If m < k n and P ∈ cAff k (R n ), then we write Q ∈ cAff m (P ) to indicate that Q ∈ cAff m (R n ) and Q ⊂ P .Further, let M ⊂ R n , 2 k n, P ∈ cAff k (R n ), and Q ∈ cAff k−1 (P ). We say that Q is a locally supporting hyperplane of M in the subspace P and write Q ∈ locTan P (M ) if there exist a point x ∈ P ∩ M and a neighborhood O(x) ⊂ P such that Q is a supporting hyperplane of M ∩ O(x) in P . The notion of a supporting hyperplane of M ∩ P in P is standard; if Q is such a hyperplane, we write Q ∈ Tan P (M ).In what follows, X is a real normed linear space. Along with Chebyshev sets, suns and strict suns play an important role in the paper. Recall that a subset M ⊂ X is called a sun if for each point x ∈ X \ M there exists a point y ∈ P M x (called a solar point) such that y ∈ P M [(1 − λ)y + λx] for every λ 0.(1)A set M ⊂ X is called a strict sun if for each x ∈ X \ M the set P M x of points nearest to x in M is nonempty and (1) holds for each y ∈ P M x. Clearly, a strict sun is always a sun. (The converse is not true in general.) Every Chebyshev set in a finite-dimensional space is a sun [1, Theorem 4.13]. * Supported by RFBR grant No. 02-01-00248. * * Recall that a face of a convex set C is a convex subset F ⊂ C such that if [a, b] is a straight-line segment in C and (a, b) ∩ F = ∅, then a and b belong to F .

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Section: Definitions and Auxiliary Resultsmentioning

confidence: 99%

The geometric structure of Chebyshev sets in ℓ∞(n)

Алимов

2005

Funct Anal Its Appl

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“…Частично обратный результат к теореме 8.10 установлен Кощеевым в [143]. На самом деле строгая выпуклость пространства является необходимым условием для того, чтобы замкнутоеB-связное множество было LG-множеством.…”

Section: классы солнц и связностьunclassified

“…Следующий результат для случая равномерно выпуклых пространств установлен в [143]. Структура дополнения к чебышёвским множествам и солнцам и, в частности, задача о числе компонент связности множества изучались в [3], [4], [7], [8].…”

Section: классы солнц и связностьunclassified

Связность И Солнечность В Задачах Наилучшего И Почти Наилучшего Приближения

Алимов¹,

Царьков²

2016

Uspekhi Mat. Nauk

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2016 г. январь -февраль т. 71, вып. 1 (427) УСПЕХИ МАТЕМАТИЧЕСКИХ НАУК УДК 517.982.256 Связность и солнечность в задачах наилучшего и почти наилучшего приближения А. Р. Алимов, И. Г. Царьков В обзоре рассматриваются структурные характеристики "солнц" в линейных нормированных пространствах. Особый упор делается на свойства связности и монотонной линейной связности солнц. Рассматриваются как прямые теоремы геометрической теории приближений, в которых из структурных характеристик множеств выводят их аппроксимативные свойства, так и обратные теоремы, в которых из аппроксимативных свойств множеств получают их структурные характеристики. Геометрические методы теории приближений используются для нахождения решений уравнения эйконала.Библиография: 231 название.Ключевые слова: солнце, строгое солнце, чебышёвское множество, почти наилучшее приближение, связность, бесконечная связность, монотонная линейная связность, уравнение эйконала.

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“…k>0 (see [5,Lemma 1] Koshcheev [5] proved that a bounded compact strict sun in LNS X is V-connected. It is established in Theorem 5 that the same property holds for existence sets with semicontinuous below metric projections in LNS X ~ RBR.…”

Section: K(yx)= U P(-~y+(~+i) Z(~+l) Xy)mentioning

confidence: 99%

Strict sums and semicontinuity below metric projections in linear normed spaces

Nevesenko¹

1978

Mathematical Notes of the Academy of Sciences of the USSR

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The relationships, are studied between strict suns [i] and sets with semicontinuous below metric projections, and also certain general properties of these classes of sets in linear normed spaces.There are characterized finite-dimensional normed spaces in which the class of strict suns coincides with the class of nonvacuous closed sets having semicontinuous below metric projections.It is proven that a P-connected [i] set with semicontinuous below (semicontinuous above) metric projections is V-connected [i].In this paper we shall use the notation:X is a real linear normed space (LNS), X* is

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The connectivity and approximative properties of sets in linear normed spaces

Cited by 18 publications

References 3 publications

The geometric structure of Chebyshev sets in ℓ∞(n)

The geometric structure of Chebyshev sets in ℓ∞(n)

Связность И Солнечность В Задачах Наилучшего И Почти Наилучшего Приближения

Strict sums and semicontinuity below metric projections in linear normed spaces

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