Duke Mathematical Journal

On the inverse Klain map

Lukas Parapatits and Thomas Wannerer

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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.

Article information

Source
Duke Math. J., Volume 162, Number 11 (2013), 1895-1922.

Dates
First available in Project Euclid: 8 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1375966905

Digital Object Identifier
doi:10.1215/00127094-2333971

Mathematical Reviews number (MathSciNet)
MR3090780

Zentralblatt MATH identifier
1280.52003

Subjects
Primary: 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45]
Secondary: 52B45: Dissections and valuations (Hilbert's third problem, etc.) 52A40: Inequalities and extremum problems

Citation

Parapatits, Lukas; Wannerer, Thomas. On the inverse Klain map. Duke Math. J. 162 (2013), no. 11, 1895--1922. doi:10.1215/00127094-2333971. https://projecteuclid.org/euclid.dmj/1375966905


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