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Sets with external Chebyshev layer. (English. Russian original) Zbl 0988.41019

Math. Notes 69, No. 2, 269-273 (2001); translation from Mat. Zametki 69, No. 2, 303-307 (2001).
Let \(X\) be a real Banach space and \(M\) a non-empty subset of \(X\). The set \(P_M(x)= \{y\in M:\|x-y\|=p(x,M)\}\) is called a metric projection of the point \(x\in X\). Denote by \(T(M)\) the set of points \(x\in X\) such that \(P_M(x)\) contains a single point. If \(T(M)=X\) then the set \(M\) is called a Chebyshev set. If \(\{x\in X:P(x,M) >a\}\subset T(M)\) for some \(a\geq 0\), we say that the set \(M\) has an external Chebyshev layer of value \(a\). A set of the form \(M=N\setminus\mathbb{C}\), when \(N\) is a Chebyshev set, \(C\subset \text{int} N\), \(\sup_{x\in C}P(x, bdc)\leq a\) for some \(a\geq 0\) is called \(a\) – wormed Chebyshev set. The authors describe the structure of sets with an external Chebyshev layer in finite-dimensional real Banach spaces. In Theorem 1 it is shown that in an arbitrary finite-dimensional Banach space, for every \(a\geq 0\), the classes of closed sets \(M\) with external Chebyshev layer of value \(a\) and closed a wormed – Chebyshev sets coincide. In Theorem 2 it is shown that if the space is also strictly convex then the result is valid without the assumption that, sets \(M\) are closed. An example is given to show that the restrictions on the structure of space \(X\) in Theorem 2 and properties of the set \(M\) in Theorem 1 are essential.

MSC:

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B28 Spaces of operators; tensor products; approximation properties
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