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
![[img]](https://authors.library.caltech.edu/style/images/fileicons/application_pdf.png) |
PDF
- Submitted Version
See Usage Policy. 302kB |
Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20200821-083421883
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: |
| |||||||||
| ORCID: |
| |||||||||
| 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: |
| |||||||||
| 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 | |||||||||
| DOI: | 10.1007/978-3-030-46762-3_6 | |||||||||
| 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: | 16 Nov 2021 18:39 |
Repository Staff Only: item control page
