Singular sets of convex bodies and surfaces with generalized curvatures (Q1895718)

In this paper the authors consider generalized surfaces with curvature measures and study the properties of those \(k\)-dimensional subsets \(\Sigma^k\) of such surfaces where the curvatures have positive density with respect to \(k\)-dimensional Hausdorff measure. The first result (Section 2) is that the sets \(\Sigma^k\) are \((H^k, k)\)-rectifiable, next in Section 3, in the simple case of the boundary of a convex body \(K\) in \(R^3\), given any curve \(\Gamma\) of class \(C^2\), it is obtained a formula relating the curvature of \(\Gamma\) to the density of the curvatures of \(K\) at \(H^1\)-almost all points of \(\Sigma^1\cap \Gamma\). In Section 4 the authors prove that the set \(\Sigma^1\) is a one-dimensional \(C^2\)-rectifiable set and finally, in Section 5 the surfaces whose curvatures live only on integer dimensional sets are introduced. For the convex sets in \(R^3\) whose curvatures live only on integer dimensional sets, the integral functionals depending on the curvature and the area of \(K\) and on the curvature and \(H^k\)-measure of \(\Sigma^k\) are considered, too.