On a sufficient condition for proximity

A closed subspace M in a Banach space X is called U-proximinal if it satisfies: $(1 + \rho )S \cap (S + M) \subseteq S + \varepsilon (\rho )(S \cap M)$, for some positive valued function $\varepsilon (\rho )$, $\rho > 0$, and $\varepsilon (\rho ) \to 0$ as $\rho\, \to\, 0$, where S is the closed unit ball of X. One of the important properties of this class of subspaces is that the metric projections are continuous. We show that many interesting subspaces are U-proximinal, for example, the subspaces with the 2-ball property (semi M-ideals) and certain subspaces of compact operators in the spaces of bounded linear operators.