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Sharp phase transitions in Euclidian integral geometry

Lotz, Martin and Tropp, Joel A. (2020) Sharp phase transitions in Euclidian integral geometry. ACM Technical Reports, 2020-01. California Institute of Technology , Pasadena, CA. (Unpublished)

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The intrinsic volumes of a convex body are fundamental invariants that capture information about the average volume of the projection of the convex body onto a random subspace of fixed dimension. The intrinsic volumes also play a central role in integral geometry formulas that describe how moving convex bodies interact. Recent work has demonstrated that the sequence of intrinsic volumes concentrates sharply around its centroid, which is called the central intrinsic volume. The purpose of this paper is to derive finer concentration inequalities for the intrinsic volumes and related sequences. These concentration results have striking implications for high-dimensional integral geometry. In particular, they uncover new phase transitions in formulas for random projections, rotation means, random slicing, and the kinematic formula. In each case, the location of the phase transition is determined by reducing each convex body to a single summary parameter.

Item Type:Report or Paper (Technical Report)
Lotz, Martin0000-0001-8500-864X
Tropp, Joel A.0000-0003-1024-1791
Alternate Title:Sharp phase transitions in integral geometry
Additional Information:5 May 2020. Revised 15 December 2021. This paper realizes a vision that was put forth by our colleague Michael McCoy in 2013. The project turned out to be more challenging than anticipated. The authors would like to thank Jiajie Chen and De Huang for their insights on the concentration of information inequality. Franck Barthe, Arnaud Marsiglietti, Michael McCoy, James Melbourne, Ivan Nourdin, Giovanni Peccati, Rolf Schneider, and Ramon Van Handel provided valuable feedback on the first draft of this paper. MAL would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “Approximation, Sampling and Compression in Data Science”, when work on this paper was undertaken. This work was supported by EPSRC grant number EP/R014604/1. JAT gratefully acknowledges funding from ONR awards N00014-11-1002, N00014-17-12146, and N00014-18-12363. He would like to thank his family for support during these trying times.
Group:Applied & Computational Mathematics
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)EP/R014604/1
Office of Naval Research (ONR)N00014-11-1002
Office of Naval Research (ONR)N00014-17-12146
Office of Naval Research (ONR)N00014-18-12363
Subject Keywords:Concentration inequality; convex body; intrinsic volumes; integral geometry; mixed volume; phase transition, quermassintegral
Series Name:ACM Technical Reports
Issue or Number:2020-01
Classification Code:Mathematics Subject Classification. Primary: 52A22, 52A39. Secondary: 52A23, 52A20, 60D05.
Record Number:CaltechAUTHORS:20220829-181401723
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:116601
Deposited By: Katherine Johnson
Deposited On:29 Aug 2022 19:24
Last Modified:09 Sep 2022 16:29

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