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Vol. 26, pp. 189–206. Am. Math. Soc. Providence, RI (1984) 22. : Geometry of log-concave functions and measures. Geom. Dedicata 112, 169–182 (2005) 23. : Uniform almost sub-gaussian estimates for linear functionals on convex sets. To appear in Algebra Anal. (St. Petersb. Math. ) 24. : Average volume of sections of star bodies. In: Geometric Aspects of Functional Analysis, Israel Seminar (1996–00). Lect. , vol. 1745, pp. 119–146. Springer, Berlin (2000) 25. : Spectral gap, logarithmic Sobolev constant, and geometric bounds.

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.

Anal. 16(5), 1021– 1049 (2006) 41. : The volume of convex bodies and Banach space geometry. In: Cambridge Tracts in Mathematics, vol. 94. Cambridge University Press, Cambridge (1989) 42. : Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. 32, 301–316 (1971) 43. : On logarithmic concave measures and functions. Acta Sci. Math. 34, 335–343 (1973) 44. : Randomized Central Limit Theorems – Probabilisitic and Geometric Aspects. PhD dissertation. Tel-Aviv University (2001) 45.

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A central limit theorem for convex sets by Klartag B.


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