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Log-concavity properties of Minkowski valuations


Authors: Astrid Berg, Lukas Parapatits, Franz E. Schuster and Manuel Weberndorfer; With an appendix by Semyon Alesker
Journal: Trans. Amer. Math. Soc. 370 (2018), 5245-5277
MSC (2010): Primary 52A38, 52B45
DOI: https://doi.org/10.1090/tran/7434
Published electronically: March 21, 2018
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Abstract: New Orlicz Brunn-Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are explored. It is shown that both lead to the same class of Minkowski valuations for which these inequalities hold. An appendix by
Semyon Alesker contains the proof of a new description of generalized translation invariant valuations.


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Additional Information

Astrid Berg
Affiliation: Vienna University of Technology, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstrasse 8–10, 1040 Vienna, Austria
Email: aberg@posteo.net

Lukas Parapatits
Affiliation: Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092 Zurich, Switzerland
Email: lukas.parapatits@math.ethz.ch

Franz E. Schuster
Affiliation: Vienna University of Technology, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstrasse 8–10, 1040 Vienna, Austria
Email: franz.schuster@tuwien.ac.at

Manuel Weberndorfer
Affiliation: Vienna University of Technology, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstrasse 8–10, 1040 Vienna, Austria
Email: m.weberndorfer@gmail.com

Semyon Alesker
Affiliation: Department of Mathematics, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel

DOI: https://doi.org/10.1090/tran/7434
Received by editor(s): September 2, 2015
Received by editor(s) in revised form: February 11, 2017
Published electronically: March 21, 2018
Additional Notes: The work of the first, second, and third authors was supported by the European Research Council (ERC) within the project “Isoperimetric Inequalities and Integral Geometry”, Project number: 306445. The second author was also supported by the ETH Zurich Postdoctoral Fellowship Program and the Marie Curie Actions for People COFUND Program.
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