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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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