An intersection property of balls and relations with M-ideals (Q1822301)

We investigate the following intersection property of balls in a Banach space X: for every \(\epsilon >0\) there is a finite family \((x_ i)\) in the open unit ball of X such that \(\| x-x_ i\| \leq 1\) for all i implies \(\| x\| \leq \epsilon\). Although this property and its negation have no nice stability behaviour we can characterize their isomorphic versions. It is known that a Banach space which can be a proper M-ideal fails this property. We construct a counterexample to show that the converse is not true. Applications are given to ultraproducts of \(L^ 1\)-preduals, to strong extreme points, and to the geometry of spaces of compact operators.