A remark on mixed curvature measures for sets with positive reach
For any two subsets \(X,Y\subseteq \mathbb{R}^d\) with positive reach and for \(r,s\in \{0,1, \dots, d-1\}\), \(r+s\geq d\), the mixed curvature measures \(C_{r,s} (X,Y;\cdot)\) have been defined by the authors in [Geom. Dedicata 57, 259-283 (1995; Zbl 0844.53050)], where, under a non-osculating condition, a translative integral formula involving \(C_{r,s} (X,Y;\cdot)\) was proved. In the paper under review the authors show, through an interesting example, that the non-osculating condition does not ensure the locally bounded variation of the mixed curvature measures. In order to have \(C_{r,s} (X,Y;\cdot)\) well defined they assume that certain nonnegative measures \(|\overline C |_{r,s} (X,Y;\cdot)\), \(0\leq r\), \(s\leq d-1\), \(r+s\geq d\) are locally finite, and look for sufficient conditions for the property. This is always satisfied if \(d\leq 3\), as well as for any pair of convex sets. Moreover, for any \(X,Y \subseteq \mathbb{R}^d\) with positive reach, \(|\overline C|_{r,s} (X,\rho Y; \cdot)\) is locally finite for \(\vartheta_d\)-almost all rotations \(\rho\in SO(d)\), where \(\vartheta_d\) is the rotation invariant probability measure on \(SO(d)\). Further, for any \(X\subseteq\mathbb{R}^d\) with positive reach and locally bounded tangential projections, and for any \(L\) in the Grassmannian \(G(d,j)\) of \(j\)-dimensional linear subspaces in \(\mathbb{R}^d\) \((0<j\leq d)\), the pair \(X,L\) satisfies the non-osculating condition and \(|\overline C|_{r,s} (X,L;\cdot)\) is locally finite.
scientific article; zbMATH DE number 1784600