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Concentration of the Intrinsic Volumes of a Convex Body

Lotz, Martin and McCoy, Michael B. and Nourdin, Ivan and Peccati, Giovanni and Tropp, Joel A. (2020) Concentration of the Intrinsic Volumes of a Convex Body. In: Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2017-2019 Volume II. Lecture Notes in Mathematics. No.2266. Springer , Cham, pp. 139-167. ISBN 978-3-030-46761-6. https://resolver.caltech.edu/CaltechAUTHORS:20200821-083421883

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Abstract

The intrinsic volumes are measures of the content of a convex body. This paper applies probabilistic and information-theoretic methods to study the sequence of intrinsic volumes. The main result states that the intrinsic volume sequence concentrates sharply around a specific index, called the central intrinsic volume. Furthermore, among all convex bodies whose central intrinsic volume is fixed, an appropriately scaled cube has the intrinsic volume sequence with maximum entropy.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/978-3-030-46762-3_6DOIArticle
ORCID:
AuthorORCID
Tropp, Joel A.0000-0003-1024-1791
Additional Information:© 2020 Springer Nature Switzerland AG. First Online: 09 July 2020. We are grateful to Emmanuel Milman for directing us to the literature on concentration of information. Dennis Amelunxen, Sergey Bobkov, and Michel Ledoux also gave feedback at an early stage of this project. Ramon Van Handel provided valuable comments and citations, including the fact that ULC sequences concentrate. We thank the anonymous referee for a careful reading and constructive remarks. Parts of this research were completed at Luxembourg University and at the Institute for Mathematics and its Applications (IMA) at the University of Minnesota. Giovanni Peccati is supported by the internal research project STARS (R-AGR-0502-10) at Luxembourg University. Joel A. Tropp gratefully acknowledges support from ONR award N00014-11-1002 and the Gordon and Betty Moore Foundation.
Funders:
Funding AgencyGrant Number
Luxembourg UniversityR-AGR-0502-10
Office of Naval Research (ONR)N00014-11-1002
Gordon and Betty Moore FoundationUNSPECIFIED
Subject Keywords:Alexandrov–Fenchel inequality; Concentration; Convex body; Entropy; Information theory; Intrinsic volume; Log-concave distribution; Quermassintegral; Ultra-log-concave sequence
Series Name:Lecture Notes in Mathematics
Issue or Number:2266
Record Number:CaltechAUTHORS:20200821-083421883
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200821-083421883
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:105053
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:21 Aug 2020 15:47
Last Modified:21 Aug 2020 15:47

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