On Certain Diameters of Bounded Sets

Aref Kamal

Abstract: In this paper 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.

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