Absolute continuity for curvature measures of convex sets. II (Q1818943)

In continuation of his previous work [Math. Nachr. 195, 139-158 (1998; Zbl 0938.52003)], the author studies the absolute continuity of the \(r\)th curvature measure \(C_r(K,\cdot)\) of a convex body \(K\subset \mathbb{R}^d\) with interior points (w.r.t. \(C_{d-1}(K,\cdot)\), the \((d-1)\)-dimensional Hausdorff measure on the boundary of \(K)\). Based on a representation of the singular part of \(C_r(K,\cdot)\) from Part I, the absolute continuity of \(C_r(K,\cdot)\) on a Borel set \(\beta\subset\mathbb{R}^d\) is expressed in terms of generalized curvatures on the unit normal bundle of \(K\). For \(r=0\), the absolute continuity is then shown to be equivalent to a supporting property from inside by \(d\)-dimensional balls. This result is generalized to \(r\in\{1, \dots,d-2\}\) with the help of the Crofton formula for curvature measures, the corresponding supporting property from inside by \((d-r)\)-dimensional balls is then required for (almost) all sections \(K\cap E\) of \(K\) with affine \((d-r)\)-flats \(E\). A formulation in terms of touching flats \(E\) is also given. The main tool for the application of the integral geometric results is a theorem, which relates the absolute continuity of curvature measures of \(K\) with the corresponding properties of sections \(K\cap E\). The interesting paper contains also various analogous results for surface area measures.