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

By Klartag B.

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Euclidean structure in finite dimensional normed spaces. In: Handbook of the Geometry of Banach Spaces, Vol. I, pp. 707–779. NorthHolland, Amsterdam (2001) 16. : Gaussian processes and almost spherical sections of convex bodies. Ann. Probab. 16(1), 180–188 (1988) 17. : On Milman’s inequality and random subspaces which escape through a mesh in Rn . In: Geometric Aspects of Functional Analysis, Israel Seminar (1986–87). Lect. , vol. 1317, pp. 84–106. Springer, Berlin (1988) 18. : Dimension, nonlinear spectra and width.

Rn |ξ|e−2π α(n+1) 2 Rn 2 n −α |ξ|2 |ξ|e−2π dξ 2 |ξ|2 dξ (23) Since x ∈ Rn and U ∈ O(n) are arbitrary, the lemma follows from (23) by the Fourier inversion formula. 1 in order to show that a typical marginal is very close, in the total-variation metric, to a spherically-symmetric concentrated distribution. A random vector X in Rn has a spherically-symmetric distribution if Prob{X ∈ U(A)} = Prob{X ∈ A} for any measurable set A ⊂ Rn and an orthogonal transformation U ∈ O(n). 1. 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.

This completes the proof of (10). A central limit theorem for convex sets 121 For x ∈ Rn and δ > 0 denote B(x, δ) = {y ∈ Rn ; |y − x| ≤ δ}. Fix x ∈ K 0 such that B(x, n −3α ) ⊂ K 0 . Then for any y ∈ B(x, n −10α ) we have y ∈ K and hence |∇ψ(y)| ≤ n 5α , by (10). Consequently, |ψ(y) − ψ(x)| ≤ n 5α |x − y| ≤ n −5α for all y ∈ B(x, n −10α ). Recalling that f = eψ , we obtain | f(y) − f(x)| ≤ 2n −5α f(x) for all y ∈ B(x, n −10α ). 1(i). According to [14, Theorem 4], sup f ≤ en f(0) ≤ e(α+1)n f(x), (15) since x ∈ K 0 .