Measure-valued valuations and mixed curvature measures of convex bodies (Q1303809)

A valuation \(\varphi\) on the space \({\mathcal K}^d\) of convex bodies (compact convex sets) in \(\mathbb{R}^d\), taking values in some real topological vector space \(X\), satisfies \(\varphi(K\cup L)+ \varphi(K\cap L)= \varphi(K) + \varphi(L)\) whenever \(K,L,K\cup L\in {\mathcal K}^d\). Then \(\varphi\) is translation covariant if it permits polynomial expansions under translation, and continuity is with respect to the Hausdorff metric. Here, the authors show that a measure-valued such valuation permits polynomial expansions in Minkowski linear combinations of convex bodies, of the form \(\lambda_1 K_1+ \cdots \lambda_k K_k\), with \(K_1, \dots,K_k \in{\mathcal K}^d\) and \(\lambda_1, \dots, \lambda_k\geq 0\). They then apply their results to obtain mixed curvature measures of convex bodies in general position, and further to generalize a translative integral geometric formula for non-intersecting convex bodies. Finally, they describe relative support measures of a convex body, which generalize support (generalized curvature) measures introduced by \textit{R. Schneider} [`Convex bodies: the Brunn-Minkowski theory', Cambridge Univ. Press (1993; Zbl 0798.52001)].