A central limit theorem for convex sets by Klartag B. PDF

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The first step of the proof is to show that {x ∈ K 0 ; |∇ψ(x)| > n 5α } ⊂ K. (10) Note that f(0) > 0 by [14, Theorem 4], and hence f(x) > 0 for all x ∈ K 0 . Consequently, ψ is finite on K 0 , and ∇ψ is well-defined on K 0 . In order to prove (10), let us pick x ∈ K 0 such that |∇ψ(x)| > n 5α . Set θ = ∇ϕ(x)/|∇ϕ(x)|. To prove (10), it suffices to show that x − n −4α θ ∈ K 0 , by the definition of K . According to the definition of K 0 , it is enough to prove that f(x − n −4α θ) < e−αn f(0). (11) We thus focus on proving (11).

There exist universal constants C1 , c, C > 0 for which the following holds: Let n ≥ 2 be an integer, and let f : Rn → [0, ∞) be an isotropic, log-concave function. Let X be a random vector in Rn with density f . Assume that sup Mf (θ, t) ≤ e−C1 n log n + inf Mf (θ, t) for all t ∈ R. θ∈S n−1 θ∈S n−1 (24) A central limit theorem for convex sets 123 Then there exists a random vector Y in Rn such that (i) dTV (X, Y ) ≤ C/n 10. (ii) Y has a spherically-symmetric distribution. √ √ 2 (iii) Prob{| |Y | − n | ≥ ε n} ≤ Ce−cε n for any 0 ≤ ε ≤ 1.

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A central limit theorem for convex sets by Klartag B.

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