Star valuations and dual mixed volumes
The Brunn-Minkowski theory and its dual regarded to convex bodies and to star-shaped sets, is extended to a broad class of bodies defined as \(L^p\)-stars. A star-shaped set \(K\subseteq \mathbb{R}^n\) is an \(L^p\)-star if the radial function \(p_K\) of \(K\) is an \(L^p\) function on \(S^{n-1}\); two \(L^p\)-stars \(K,L\) are defined to be equal whenever \(p_K=p_L\) almost everywhere on \(S^{n-1}\). If \(p_K\) is a continuous function on \(S^{n-1}\), then \(K\) is called a star body. Then a definition is given for the dual Hausdorff topology on \(L^p\)-stars which permits a continuous action of the rotation group \(SO(p)\) on the space of \(L^p\)-stars. Definitions are given for valuations of \(L^p\)-stars; the paper presents an important classification of all continuous star valuations that are homogeneous with respect to delation.
scientific article; zbMATH DE number 934292