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Journal of Mathematical Sciences

, Volume 238, Issue 4, pp 537–559 | Cite as

Gaussian Convex Bodies: a Nonasymptotic Approach

  • G. PaourisEmail author
  • P. Pivovarov
  • P. Valettas
Article
  • 6 Downloads

We study linear images of a symmetric convex body C ⊆ ℝN under an n × N Gaussian random matrix G, where Nn. Special cases include common models of Gaussian random polytopes and zonotopes. We focus on the intrinsic volumes of GC and study the expectation, variance, small and large deviations from the mean, small ball probabilities, and higher moments. We discuss how the geometry of C, quantified through several different global parameters, affects such concentration properties. When n = 1, G is simply a 1 × N row vector, and our analysis reduces to Gaussian concentration for norms. For matrices of higher rank and for natural families of convex bodies CN ⊆ ℝN, with N → ∞, we obtain new asymptotic results and take first steps to compare with the asymptotic theory.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Texas A&M UniversityCollege StationUSA
  2. 2.University of MissouriColumbiaUSA

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