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On the volume of sections of a convex body by cones


Authors: Matthieu Fradelizi, Mathieu Meyer and Vlad Yaskin
Journal: Proc. Amer. Math. Soc. 145 (2017), 3153-3164
MSC (2010): Primary 52A20, 52A40
DOI: https://doi.org/10.1090/proc/13457
Published electronically: January 23, 2017
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Abstract: We prove that in small codimensions, the sections of a convex body in $ \mathbb{R}^n$ through its centroid are quite symmetric with respect to volume. As a consequence of our estimates we give a positive answer to a problem posed by M. Meyer and S. Reisner regarding convex intersection bodies.


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Matthieu Fradelizi
Affiliation: Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées UMR 8050, UPEMLV, UPEC, CNRS F-77454, Marne-la-Vallée, France
Email: matthieu.fradelizi@u-pem.fr

Mathieu Meyer
Affiliation: Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées UMR 8050, UPEMLV, UPEC, CNRS F-77454, Marne-la-Vallée, France
Email: mathieu.meyer@u-pem.fr

Vlad Yaskin
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: yaskin@ualberta.ca

DOI: https://doi.org/10.1090/proc/13457
Keywords: Convex body, section, $s$-concave function, centroid
Received by editor(s): April 12, 2016
Received by editor(s) in revised form: August 29, 2016
Published electronically: January 23, 2017
Additional Notes: The third author was supported in part by NSERC. Part of this work was done when the third author was visiting Université Paris-Est Marne-la-Vallée. He is grateful for its hospitality
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2017 American Mathematical Society