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.



Kinematic formula for Euler characteristic
Yes, this is called the principal kinematic formula:
$$\int \chi(K \cap gL)\, dg = \sum\_{k=0}^n c\_{nk} V\_k(K) V\_{n-k}(L),$$
where $V\_i$ are the intrinsic volumes, and $c\_{nk}$ certain constants. See e.g. Section 4.4 in


*Schneider, Rolf*, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and Its Applications. 44. Cambridge: Cambridge University Press. xiii, 490 p. (1993). [ZBL0798.52001](https://zbmath.org/?q=an:0798.52001).


At the end of that section there are historical references.


Note that if $L$ is a ball of radius $r$, then there is no dependence on the "rotational part" of $g$, so one integrates over translations only, and the formula reduces to the Steiner formula for the volume of an $r$-neighborhood of $K$.


