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 .
