Hessian measures of semi-convex functions and applications to support measures of convex bodies
The paper contains deep results on support measures (thus, in particular, on curvature and surface area measures) of convex bodies and, more generally, of sets with positive reach. \textit{R. Schneider} proved [see Theorem 4.6.5 in his book ``Convex bodies: The Brunn-Minkowski theory'' (1993; Zbl 0798.52001)] that the \(j\)th curvature measure \(C_j(K,\cdot)\) (the \(j\)th surface area measure \(S_j(K,\cdot)\)) of a convex body \(K\) is absolutely continuous with resp. to the Hausdorff measure of a suitable dimension. The authors obtain this result as a consequence of Theorem 0.1, which states that for every convex body \(K \subset \text{R}^d\) and \(j \in \{0,\dots ,d-1\}\), if \(\alpha \subset \text{bd} K\) is a Borel set with \(\sigma\)-finite \(j\)-dimensional Hausdorff measure, then \[ {d-1 \choose j} C_j(K,\alpha)= \int_\alpha H^{d-1-j}(N(K,x)\cap S^{d-1})d{\mathcal H}^j(x), \] where \(N(K,x)\) is the normal cone of \(K\) at \(x\). Furthermore, a similar assertion holds for surface area measures. They study Hessian measures \(F_k\) of semi-convex functions (the notion introduced in their earlier paper). These measures are analogues of the support measures for sets with positive reach: they are defined as coefficients in a local Steiner type formula ((4) p. 14) and satisfy a Crofton type formula (Theorem 2.1). Moreover, the measure \(F_k(u,\cdot)\) of a semi-continuous function \(u\) is absolutely continuous with resp. to the Hausdorff measure \({\mathcal H}^k\) (Theorem 3.1). Theorem 5.4 describes a simple connection between the support measures of a set \(X\) with positive reach and the Hessian measures of its distance function \(d_X\). Similarly, Corollary 5.9 describes a connection between the support measures of a convex body \(K\) and the Hessian measures of its support function \(h_K\). Using these theorems (Theorem 5.4 and Corollary 5.9) the authors deduce results on support measures of a set \(X\) with positive reach from corresponding results on the Hessian measures of \(d_X\) (Theorem 5.5); similarly, they deduce results on support measures of a convex body \(K\) from corresponding results on the Hessian measures of \(h_K\) (Theorem 5.11). Theorem 0.1 is a direct consequence of 5.5 and 5.11.
scientific article; zbMATH DE number 1423147