zbMATH — the first resource for mathematics

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.

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