  The intrinsic volumes of a convex body are fundamental invariants that
capture information about the average volume of the projection of the convex
body onto a random subspace of fixed dimension. The intrinsic volumes also play
a central role in integral geometry formulas that describe how moving convex
bodies interact. Recent work has demonstrated that the sequence of intrinsic
volumes concentrates sharply around its centroid, which is called the central
intrinsic volume. The purpose of this paper is to derive finer concentration
inequalities for the intrinsic volumes and related sequences. These
concentration results have striking implications for high-dimensional integral
geometry. In particular, they uncover new phase transitions in formulas for
random projections, rotation means, random slicing, and the kinematic formula.
In each case, the location of the phase transition is determined by reducing
each convex body to a single summary parameter.
