Curvature Measures of Singular Sets

The book describes how curvature measures can be introduced for certain classes of sets with singularities in Euclidean spaces. Its focus lies on sets with positive reach and some extensions, which include the classical polyconvex sets and piecewise smooth submanifolds as special cases. The measures under consideration form a complete system of certain Euclidean invariants. Techniques of geometric measure theory, in particular, rectifiable currents are applied, and some important integral-geometric formulas are derived. Moreover, an approach to curvatures for a class of fractals is presented, which uses approximation by the rescaled curvature measures of small neighborhoods. The book collects results published during the last few decades in a nearly comprehensive way.

- Background from Geometric Measure Theory. - Background from Convex Geometry. - Background from Differential Geometry and Topology. - Sets with Positive Reach. - Unions of Sets with Positive Reach. - Integral Geometric Formulae. - Approximation of Curvatures. - Characterization Theorems. - Extensions of Curvature Measures to Larger set Classes. - Fractal Versions of Curvatures.