Curvature Measures of Singular Sets

The book describes how curvature measures can be introduced for certain classes of sets withsingularities 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 specialcases. The measures under consideration form a complete system of certain Euclideaninvariants. Techniques of geometric measure theory, in particular, rectifiable currents areapplied, and some important integral-geometric formulas are derived. Moreover, an approach tocurvatures for a class of fractals is presented, which uses approximation by the rescaledcurvature measures of small neighborhoods. The book collects results published during the lastfew decades in a nearly comprehensive way.