On the definition of total absolute curvatures in integral geometry (Q1271934)

The author proves that the total absolute curvatures of L. A. Santaló for immersed manifolds in Euclidean space are well-defined. These quantities are given as follows. Let \(M\) be an \(n\)-dimensional manifold immersed in \({\mathbb R}^{N+n}\) and fix \(1\leq r\leq n\). For each \((n+N-r)\)-plane \(L\subset{\mathbb R}^{n+N}\), let \(\Gamma_L\) denote the set of all \(r\)-planes contained in some \(T_pM\) and orthogonal to \(L\). Then the total absolute curvature \(K_r(M)\) is the average of the Hausdorff measures of \(\Gamma_L\cap L\) over all \(L\) in the Grassmannian \(G_{n+N-r,n+N}\). There is a similar definition for \(r\geq n\). The central point of this paper is that for almost all \(L\in G_{n+N-r,n+N}\), the set \(\Gamma_L\cap L\) is almost everywhere an immersed manifold of dimension \((n-rN)\).