On the inverse Klain map

Abstract

The continuity of the inverse Klain map is investigated and the class of centrally symmetric convex bodies at which every valuation depends continuously on its Klain function is characterized. Among several applications, it is shown that McMullen's decomposition is not possible in the class of translation-invariant, continuous, positive valuations. This implies that there exists no McMullen decomposition for translation-invariant, continuous Minkowski valuations, which solves a problem first posed by Schneider and Schuster.

21 pages; typos fixed

References in corpus (3)

• Crofton measures and Minkowski valuations
• Integral geometry of complex space forms
• SL(n)-Contravariant $L_p$-Minkowski Valuations

Cited by in corpus (5)

• The Hadwiger theorem on convex functions, I
• Minkowski valuations in a 2-dimensional complex vector space
• Minkowski additive operators under volume constraints
• A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions
• Aleksandrov-Fenchel inequalities for unitary valuations of degree 2 and 3