Invariant valuations on spherical star sets (Q2707492)

\textbf{P.McMullen (London)}: In Adv. Math. 121, 80-101 (1996; Zbl 0858.52003) and Adv. Math. 125, 95-113 (1997; Zbl 0889.52007), \textit{D. A. Klain} obtained characterization results about continuous rotation invariant valuations on euclidean star shaped sets, and corresponding dual volumes, which extend Hadwiger's famous characterization of the quermassintegrals (intrinsic volumes), although there are now more such valuations than just the quermass integrals.NEWLINENEWLINENEWLINEIn this paper, the authors extend these results to certain classes of star shaped spherical sets, with the valuations again being continuous and rotation invariant. A particular class of these valuations enables sections of star sets to be studied, with consequent dual Kubota integral formulae and Funk section theorems.NEWLINENEWLINENEWLINE\textbf{Maria Moszyńska (Warszawa)}: The authors define and investigate spherical star-shaped sets and spherical star bodies. A spherical star-shaped set (i.e. star shaped at some point \(p\) subset of the unit \(n\)-sphere \(S^n\)) is determined by its spherical radial function; star body is a star-shaped set whose radial function is continuous.NEWLINENEWLINE These notions are natural counterparts of the corresponding euclidean notions.NEWLINENEWLINEThe authors develop theory of continuous and rotation invariant valuations on the class of spherical star bodies. In particular, they define spherical dual volumes, which are analogues of dual volumes of star bodies in \({\mathbb R}^n\) in the sense of Lutwak. The main results are Theorems 18 and 19. The first one in an analogue of the Kubota formula. The second is an analogue of the generalized Funk section theorem (compare R.Gardner, Geometric Tomography, Theorem 2.6); it says that (as in euclidean space) a centered spherical star-shaped set is uniquely determined by the dual volumes of its sections.NEWLINENEWLINEFor the entire collection see [Zbl 0948.00038].