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Geometric and Functional Analysis

, Volume 27, Issue 3, pp 466–488 | Cite as

Minkowski Endomorphisms

  • Felix DorrekEmail author
Open Access
Article

Abstract

Several open problems concerning Minkowski endomorphisms and Minkowski valuations are solved. More precisely, it is proved that all Minkowski endomorphisms are uniformly continuous but that there exist Minkowski endomorphisms that are not weakly-monotone. This answers questions posed repeatedly by Kiderlen (Trans Am Math Soc 358:5539–5564, 2006), Schneider (Convex bodies: the Brunn–Minkowski theory. Second expanded edition, encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge, 2014) and Schuster (Trans Am Math Soc 359:5567–5591, 2007). Furthermore, a recent representation result for Minkowski valuations by Schuster and Wannerer is improved under additional homogeneity assumptions. Also a question related to the structure of Minkowski endomorphisms by the same authors is answered. Finally, it is shown that there exists no McMullen decomposition in the class of continuous, even, SO(n)-equivariant and translation invariant Minkowski valuations extending a result by Parapatits and Wannerer (Duke Math J 162:1895–1922, 2013).

Notes

Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). The work of the author was supported by the European Research Council (ERC), Project Number: 306445, and the Austrian Science Fund (FWF), Project Number: Y603-N26.

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Vienna University of Technology, Institute of Discrete Mathematics and GeometryViennaAustria

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