Boundaries of Asplund spaces

We study the relationship between the classical combinatorial inequalities of Simons and the more recent (I)-property of Fonf and Lindenstrauss. We obtain a characterization of strong boundaries for Asplund spaces using the new concept of finitely self-predictable set. Strong properties for w∗-K-analytic boundaries are established as well as a sup–lim sup theorem for Baire maps.