Kinematic formulas in convex geometry for non-compact groups

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.