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Let $(X,d)$ be a metric space and let $$K(X)=\lbrace Y\subset X\colon Y\text{ is non-empty and compact}\rbrace.$$ Endow $K$ with the Hausdorff metric (which is the natural metric on this space, see )

Is there some sort of relation between the Hausdorff (or ball) measure of non-compactness $\alpha$ and the Hausdorff metric on $K$, that is, is there an identity of the form $\alpha(A)=d_H(A,K(X))$? See http://en.wikipedia.org/wiki/Measure_of_non-compactness and http://en.wikipedia.org/wiki/Hausdorff_distance for the relevant definitions.

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    Could you clarify please: you never introduced the symbol $d_H$ formally before using it; if $d_H$ denotes the Hausdorff metric, then I'm confused. Indeed, $K(X)$ is a subset of the power set of $X$, while $A$ (I presume) is a subset of $X$. The question as stated is very confusing.2012-03-22

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