Concentration of the Intrinsic Volumes of a Convex Body

Creators: Lotz, Martin; McCoy, Michael B.; Nourdin, Ivan; Peccati, Giovanni; Tropp, Joel A.

Others:: Klartag, Bo'az; Milman, Emanuel

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.

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.

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