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On certain diameters of bounded sets. (English) Zbl 0821.46012
Summary: We prove that if the balanced convex closed subset \(A\) of the normed linear space \(X\) has a certain property called the property \(P_ 0\), then the Gelfand \(n\)-width \(d^ n(A, X)\) is attained. If \(A\) is a balanced and compact subset of \(X\), then the Bernstein \(n\)-width \(b^ n(A, X)\) is attained, and if \(A\) is a subset of the dual space \(X^*\), and \(A\) contains a ball \(B(0, r)\) of positive radius, then the linear \(n\)-width \(\delta_ n(A, X^*)\) is attained.
It is also shown that if \(X\) has a certain property called the property \(P_ 1\), then the compact width \(a(A, X)\) is attained.

MSC:
46B20 Geometry and structure of normed linear spaces
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