  The convex hull generated by the restriction to the unit ball of a stationary
Poisson point process in the $d$-dimensional Euclidean space is considered. By
establishing sharp bounds on cumulants, exponential estimates for large
deviation probabilities are derived and the relative error in the central limit
theorem on a logarithmic scale is investigated for a large class of key
geometric characteristics. This includes the number of lower-dimensional faces
and the intrinsic volumes of the random polytopes. Furthermore, moderate
deviation principles for the spatial empirical measures induced by these
functionals are also established using the method of cumulants. The results are
applied to deduce, by duality, fine probabilistic estimates and moderate
deviation principles for combinatorial parameters of a class of zero cells
associated with Poisson hyperplane mosaics. As a special case this comprises
the typical Poisson-Voronoi cell conditioned on having large inradius.
