Intrinsic Volumes of Polyhedral Cones: A Combinatorial Perspective

The theory of intrinsic volumes of convex cones has recently found striking applications in areas such as convex optimization and compressive sensing. This article provides a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Direct derivations of the general Steiner formula, the conic analogues of the Brianchon–Gram–Euler and the Gauss–Bonnet relations, and the principal kinematic formula are given. In addition, a connection between the characteristic polynomial of a hyperplane arrangement and the intrinsic volumes of the regions of the arrangement, due to Klivans and Swartz, is generalized and some applications are presented.