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The recently introduced Lipschitz-Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a K\"unneth-type formula for Lipschitz-Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms., Comment: 25 pages

We explore the notion of Santal\'o point for the Holmes-Thompson boundary area of a convex body in a normed space. In the case where the norm is $C^1$, and in the case where unit ball and convex body coincide, we prove existence and uniqueness. When the normed space has a smooth positively curved unit ball, we exhibit a dual Santal\'o point expressed as an average of centroids of projections of the dual body., Comment: 21 pages, 2 figures. We have added a new result to the paper showing that our previous uniqueness results hold without any regularity assumptions in case the two convex bodies B and K coincide. This is the content of the new Theorem 1.2

The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric $(0,2)$-tensors. More precisely, we construct a family of generalized curvature measures attached to such manifolds, extending the Riemannian Lipschitz-Killing curvature measures introduced by Federer. We then show that they behave naturally under isometric immersions, in particular they do not depend on the ambient signature. Consequently, we extend Theorema Egregium to surfaces equipped with a generic metric of changing signature, and more generally, establish the existence as distributions of intrinsically defined Lipschitz-Killing curvatures for such manifolds of arbitrary dimension. This includes in particular the scalar curvature and the Chern-Gauss-Bonnet integrand. Finally, we deduce a Chern-Gauss-Bonnet theorem for pseudo-Riemannian manifolds with generic boundary., Minor improvements to exposition following referee's report. To appear in Crelle's Journal. 45 pages

We introduce a Mobius invariant energy associated to planar domains, as well as a generalization to space curves. This generalization is a Mobius version of Banchoff-Pohl's notion of area enclosed by a space curve. A relation with Gauss-Bonnet theorems for complete surfaces in hyperbolic space is also described.

We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives the Gauss curvature integral in terms of the Euler characteristic, and some hermitian intrinsic volumes. The other one, which is shorter, involves the measure of complex hyperplanes meeting the domain. As a tool, we obtain variation formulas in integral geometry of complex space forms., 24 pages, to appear in Israel Journal of Mathematics