The approximate hyperplane series property and related properties

Abstract

We show that the approximate hyperplane series property consequence, we obtain that the class of spaces $Y$ such that the pair $({\ell }_1,Y)$ has the Bishop–Phelps–Bollobás property for operators is stable under finite ${\ell }_p$-sums for $1\le p<\infty$. We also deduce that every Banach space of dimension at least $2$ can be equivalently renormed to have the AHSp but to fail Lindenstrauss’ property $\beta$. We also show that every infinite-dimensional Banach space admitting an equivalent strictly convex norm also admits such an equivalent norm failing the AHSp.