  We continue a line of study about some local versions of
Bishop-Phelps-Bollob\'as type properties for bounded linear operators. We
introduce and focus our attention on two of these local properties, which we
call L$_{p, o}$ and L$_{o, p}$, and we explore the relation between them and
some geometric properties of the underlying spaces, such as spaces having
strict convexity, local uniform rotundity, and property $\beta$ of
Lindenstrauss. At the end of the paper, we present a diagram comparing all the
existing Bishop-Phelps-Bollob\'as type properties with each other. Some open
questions are left throughout the article.
