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.