This book provides the foundations for geometric applications of convex
cones and presents selected examples from a wide range of topics,
including polytope theory, stochastic geometry, and Brunn-Minkowski
theory. Giving an introduction to convex cones, it describes their most
important geometric functionals, such as conic intrinsic volumes and
Grassmann angles, and develops general versions of the relevant
formulas, namely the Steiner formula and kinematic formula.
In recent years questions related to convex cones have arisen in applied
mathematics, involving, for example, properties of random cones and
their non-trivial intersections. The prerequisites for this work, such
as integral geometric formulas and results on conic intrinsic volumes,
were previously scattered throughout the literature, but no coherent
presentation was available. The present book closes this gap. It
includes several pearls from the theory of convex cones, which should be
better known.