  We present a sufficient condition for a Banach space to have the approximate
hyperplane series property (AHSP) which actually covers all known examples. We
use this property to get a stability result to vector-valued spaces of
integrable functions. On the other hand, the study of a possible
Bishop-Phelps-Bollob\'as version of a classical result of V. Zizler leads to a
new characterization of the AHSP for dual spaces in terms of $w^*$-continuous
operators and other related results.
