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

Abstract

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 .

Characterisations of Chebyshev sets in c0

“…The importance of coordinate subspaces for approximation theory was shown in [3] for X ¼ c N ðnÞ: Here we obtain similar results for c 0 : Theorem 1 states that for an approximatively compact Chebyshev set M in c 0 and for HAcAff oÀk ðc 0 Þ; kAZ þ ; if M-Ha|; then HCTðM-HÞ;…”

“…This result will be obtained as a corollary from a more general Theorem 2 below. The similar result is also true in c N ðnÞ (see [3]). …”

“…Theorem 4 characterises approximatively compact Chebyshev sets in c 0 in terms of their intersections with coordinate affine ARTICLE IN PRESS $ The work was supported by the Russian Fund for Basic Researches, project 02-01-00248. subspaces of finite codimension. Theorems 1-3 establish that an intersection of an approximatively compact Chebyshev set in c 0 with a coordinate affine subspace H of finite codimension or with a finite intersection H of coordinate half-spaces preserves its approximative properties with respect to H: Similar results for c N ðnÞ were recently obtained by the author [2,3].…”

“…The similar characterisation for Chebyshev sets in c N ðnÞ was obtained in [2,3]. The paper has the following structure.…”

“…Thus, we give the following definitions: cAff oÀk ðc 0 Þ (kAZ þ ) will denote the set of all coordinate affine subspaces of c 0 of finite codimension k; i.e., affine subspaces H which are parallel to a correspondent face F of the unit ball, codim F ¼ k: In other words, H ¼ fxAc 0 j x i 1 ¼ c 1 ; y; x i k ¼ c k g for some fixed set of indices i 1 ; y; i k and set of constants c 1 ; y; c k ; cAff k ðc 0 Þ ðkANÞ will denote the set of all coordinate affine subspaces of c 0 of finite dimension k (see [3]); i.e., cAff k ðc 0 Þ consists of affine subspaces of the following form: linfe i 1 ; y; e i k j 1pi 1 o?oi k oNg þ x; xAc 0 ; here e 1 ; e 2 ; y is the natural basis of c 0 :…”