Intrinsic Volumes of Polyhedral Cones: A combinatorial perspective

TL;DR

This paper offers a comprehensive combinatorial framework for intrinsic volumes of polyhedral cones, deriving key formulas and connecting them to hyperplane arrangements, with applications in convex optimization and related fields.

Contribution

It provides self-contained derivations of fundamental formulas and generalizes the connection between hyperplane arrangements and intrinsic volumes.

Findings

Derivation of the General Steiner formula for polyhedral cones

Extension of the connection between characteristic polynomials and intrinsic volumes

Applications to convex optimization and compressive sensing

Abstract

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.