Kinematic formulas in convex geometry for non-compact groups

Sílvia Anjos; Francisco Nascimento

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arXiv · 2504.06743

Kinematic formulas in convex geometry for non-compact groups

Sílvia Anjos·Francisco Nascimento

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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 $G \cong K \times \exp(\mathfrak{p}_0)$, 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 $\overline{G} = G \ltimes V$. Key to our approach is the introduction of a Gaussian measure on $\mathfrak{p}_0$, 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.

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Sílvia Anjos, Francisco Nascimento. 2025-04-09. Kinematic formulas in convex geometry for non-compact groups. https://arxiv.org/abs/2504.06743

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