Bad normed spaces, convexity properties, separated sets Original Research Article

Some years ago, a parameter–denoted by A1(X)A1(X)–was defined in real Banach spaces. In the same setting, several years before, a notion called QQ-convexity had been defined. Studying these two notions seems to be rather awkward and up until now this has not been done in deep. Here we indicate some properties and connections between these two parameters and some other related ones, in infinite-dimensional Banach spaces. We also consider another notion, a natural extension of QQ-convexity, and we discuss the case when A1(X)A1(X) attains its maximum value. The spaces where this happens can be considered as ”bad” since they cannot have several properties which are usually considered as nice (like uniform non-squareness or PP-convexity).