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.
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asked Sep 23, 2019 at 10:02
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$\begingroup$ 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. $\endgroup$– Ben McKayCommented Sep 23, 2019 at 10:08
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1$\begingroup$ $\chi(K \cap gL) = 1$ if $K \cap gL \ne \emptyset$ and $0$ otherwise. $\endgroup$– Ivan IzmestievCommented Sep 23, 2019 at 11:00
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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.
At the end of that section there are historical references.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 $K$.
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$\begingroup$ This wasn't the question, but this is true for sufficiently regular non convex sets as well, although finding the maximal class of subsets for which this holds is open and an active area of research $\endgroup$– alesiaCommented Sep 23, 2019 at 15:45