# Kinematic formula for Euler characteristic

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.

• I can't picture the Euler characteristic being nonconstant, over a family of convex sets. If it is constant, then this seems like a constant multiple of volume of the isometry group. Sep 23, 2019 at 10:08
• $\chi(K \cap gL) = 1$ if $K \cap gL \ne \emptyset$ and $0$ otherwise. Sep 23, 2019 at 11:00

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
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 $$K$$.