Measures on polyhedral cones: characterizations and kinematic formulas

Abstract

This paper is about conic intrinsic volumes and their associated integral geometry. We pay special attention to the biconic localizations of the conic intrinsic volumes, the so-called support measures. An analysis of these quantities has so far been confined to the PhD thesis of Stefan Glasauer (1995). We rederive the results from this thesis with novel streamlined proofs and expand them in several ways. Additionally, we introduce a new class of functionals on polyhedral cones lying between the intrinsic volumes and the well-studied f-vector, which counts the equidimensional faces of a cone, and derive a characterization and kinematic formulas for these functionals as well.