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.
Highlights
The theory of conic intrinsic volumes has a rich and varied history, with origins dating back at least to the work of Sommerville [32]
Sect. 2.3), or are available as special cases of a more sophisticated theory of spherical integral geometry [13,30,34] that treats the subject in a level of generality that is usually more than what is needed from the point of view of the above-mentioned applications
Conic polarity swaps between internal and external angles: β(F, C) = γ (F, C◦), γ (F, C) = β(F, C◦), where we use the notation F := NF C for the face of C◦, which is polar to the face F of C. This shows that any formula involving the internal and external angles of a cone C has a polar version in terms of the internal and external angles of C◦ where the roles of internal and external have been exchanged. (Some of the formulas in [25] are stated in this polar version.) Remark 2.9 The Brianchon–Gram–Euler relation [27, Thm. (1)] of a convex polytope K translates in the above notation as
Summary
The theory of conic intrinsic volumes (or solid/internal/external/Grassmann angles) has a rich and varied history, with origins dating back at least to the work of Sommerville [32] This theory has recently found renewed interest, owing to newly found connections with measure concentration and resulting applications in compressive sensing, optimization, and related fields [3,5,11,14,24]. While some of the material is classic (see, for example, [25]), we blend into the presentation a generalization of a formula of Klivans and Swartz [22], with a streamlined proof and some applications The focus of this text is on simplicity rather than generality, on finding the most natural relations between different results that may be derived in different orders from each other, and on highlighting parallels between different results.
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