Is there a formula for $\int \chi(K \cap gL) \: dg$ (where $\chi$ is Euler characteristic) analogous to the kinematic formula for $\int \mu(K \cap gL) \: dg$ (where $\mu$ is Lebesgue measure)? In both expressions $K$ and $L$ are compact convex bodies, $g$ varies over a group of isometries acting on the ambient space, and $dg$ signifies integration with respect to the Haar measure of that group.