Classification of angular curvature measures and a proof of the angularity conjecture

Abstract

In this paper angular curvature measures are investigated. Our first result is a complete classification of translation-invariant angular smooth curvature measures on $\mathbb{R}^n$. Subsequently, we use this result to show that the class of angular curvature measures on a Riemannian manifold is preserved by both the pullback by isometric immersions and the action of the Lipschitz-Killing algebra. The latter confirms the angularity conjecture formulated by A. Bernig, J.H.G. Fu, and G. Solanes.