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Lipschitz-Killing invariants. (English) Zbl 1074.53064
The authors define and characterize Lipschitz-Killing invariants for lattices of compact, sufficiently tame subsets of the Euclidean space. Their main example consists in definable subsets with respect to an $$O$$-minimal system $$\omega$$, a concept introduced previously in [L. van den Dries, Tame topology and $$O$$-minimal structures (London Mathematical Society Lecture Note Series 248, Cambridge University Press, Cambridge) (1998; Zbl 0953.03045)]. Let $$M_0$$ denote the $$\mathbb{R}$$-algebra generated by all definable isometry classes $$[Y]$$, $$Y\in\omega$$, $$Y$$ compact, subject to all relations $$[Y\cup Z]+ [Y\cap Z]= [Y]+ [Z]$$. This is the metric counterpart of the universal ring $$K_0(\omega)$$. In a similar fashion as to the Euler characteristic (with respect to the Borel-Moore homology), which defines an isomorphism $$\chi: K_0(\omega)\to \mathbb{Z}$$, the authors define a homomorphism, the so-called total Lipschitz-Killing invariant, denoted by $$\Lambda: M_0(\omega)\to \mathbb{R}[t]$$. The results they obtain are also interpreted in terms of spherical currents.

##### MSC:
 53C65 Integral geometry 14P10 Semialgebraic sets and related spaces 52A22 Random convex sets and integral geometry (aspects of convex geometry)
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