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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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