Kinematic formulas in convex geometry for non-compact groups

2025/04/09 by Anjos, Sílvia, Nascimento, Francisco
#22E46 #FOS: Mathematics #Metric Geometry (math.MG) #Primary 53C65 #Secondary 52A22

paper · doi:10.48550/arxiv.2504.06743

Abstract

We generalize classical kinematic formulas for convex bodies in a real vector space V to the setting of non-compact Lie groups admitting a Cartan decomposition. Specifically, let G be a closed linear group with Cartan decomposition G ≅ K × exp(\mathfrakp₀), where K is a maximal compact subgroup acting transitively on the unit sphere. For K-invariant continuous valuations on convex bodies, we establish an integral geometric-type formula for G = G \ltimes V. Key to our approach is the introduction of a Gaussian measure on \mathfrakp₀, which ensures convergence of the non-compact part of the integral. In the special case K = O(n), we recover a Hadwiger-type formula involving intrinsic volumes, with explicit constants c_j computed via a Weyl integration formula.

Related

• The kinematic formula in Riemannian homogeneous spaces
• Kinematic formulas for sets defined by differences of convex functions
• Kinematic formulas for area measures
• Horn problem for quasi-hermitian Lie groups
• A general Weyl-type Integration Formula for Isometric Group Actions