  Probability boxes, also known as $p$-boxes, correspond to sets of probability
distributions bounded by a pair of distribution functions. They fall into the
class of models known as imprecise probabilities. One of the central questions
related to imprecise probabilities are the intervals of values corresponding to
expectations of random variables, and especially the interval bounds. In
general, those are attained in extremal points of credal sets, which denote
convex sets of compatible probabilistic models. The aim of this paper is a
characterization and identification of extreme points corresponding to
$p$-boxes on finite domains. To accomplish this, we utilize the concept of
normal cones. In the settings of imprecise probabilities, those correspond to
sets of random variables whose extremal expectations are attained in a common
extreme point. Our main results include a characterization all possible normal
cones of $p$-boxes, their relation with extreme points, and the identification
of adjacency structure on the collection of normal cones, closely related to
the adjacency structure in the set of extreme points.
