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Published August 29, 2022 | Accepted Version
Report Open

Sharp phase transitions in Euclidian integral geometry

Abstract

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.

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.

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Created:
August 22, 2023
Modified:
January 15, 2024