Abstract
Let K N = [±G 1, . . . , ±G N ] be the absolute convex hull of N independent standard Gaussian random points in \({\mathbb R^n}\) with N ≥ n. We prove that, for any 1-Lipschitz function \({f:\mathbb R^n\rightarrow\mathbb R}\), the polytope K N satisfies the following Poincaré inequality in mean value:
where C > 0 is an absolute constant. This Poincaré inequality is the one suggested by a conjecture of Kannan, Lovász and Simonovits for general convex bodies. Moreover, we prove in mean value that the volume of the polytope K N is concentrated in a subexponential way within a thin Euclidean shell with the optimal dependence of the dimension n. An important tool of the proofs is a representation of the law of (G 1, . . . , G n ) conditioned by the event that “the convex hull of G 1, . . . , G n is a (n − 1)-face of K N ”. As an application, we also get an estimate of the number of (n − 1)-faces of the polytope K N , valid for every N ≥ n.
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Fleury, B. Poincaré inequality in mean value for Gaussian polytopes. Probab. Theory Relat. Fields 152, 141–178 (2012). https://doi.org/10.1007/s00440-010-0318-3
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DOI: https://doi.org/10.1007/s00440-010-0318-3
Mathematics Subject Classification (2000)
- 52B11
- 52B60