Kinematic formulas in convex geometry for non-compact groups
Sílvia Anjos, Francisco Nascimento
TL;DR
This work extends classical kinematic formulas from compact to non-compact groups with Cartan decomposition by integrating valuations over $\overline{G}=G\ltimes V$ with a Gaussian measure on $\mathfrak{p}_0$. The main result expresses the integral of a $K$-invariant valuation $\phi$ applied to $M\cap \overline{g}L$ as a finite combination of $K$-invariant valuations on $V$ and intrinsic volumes, with computable coefficients. In the special case $K=O(n)$, the framework recovers a Hadwiger-type formula with explicit constants $c_j$ determined via Weyl integration on symmetric matrices, enabling direct computation (e.g., $c_n=e^{n/2}$). Overall, the paper provides a principled method to obtain integral-geometric formulas in non-compact settings, offering practical coefficients and a clear pathway for concrete groups such as $O(n)$, $GL(n)$, and related Hermitian and symplectic cases. The approach broadens the scope of affine and intrinsic-volume-based techniques to a broad class of non-compact Lie groups, with potential applications in convex-geometric analysis on homogeneous spaces.
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.