Absolute continuity for curvature measures of convex sets. II
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.
scientific article; zbMATH DE number 1384887