By Klartag B.

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**Example text**

Then, sup g(tθ) ≤ e−αn log n + inf g(tθ) for all t ≥ 0. 2. , [49, p. 6]). Therefore g(ξ ˆ 1 ) − g(ξ ˆ 2 ) ≤ re−2π 2 n −α r 2 e−2αn log n when |ξ1 | = |ξ2 | = r. (22) Let x ∈ R and U ∈ O(n). From (22), n Rn 2πi (g(ξ) ˆ − g(Uξ))e ˆ x,ξ dξ ≤ e−2αn log n = e−2αn log n n ≤ e−αn log n . 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.

Therefore g(ξ ˆ 1 ) − g(ξ ˆ 2 ) ≤ re−2π 2 n −α r 2 e−2αn log n when |ξ1 | = |ξ2 | = r. (22) Let x ∈ R and U ∈ O(n). From (22), n Rn 2πi (g(ξ) ˆ − g(Uξ))e ˆ x,ξ dξ ≤ e−2αn log n = e−2αn log n n ≤ e−αn log n . 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.

The geometry of logconcave functions and sampling algorithms. Random Struct. Algorithms (published online) 30. : The central limit problem for random vectors with symmetries. Preprint. 31. : On gaussian marginals of uniformly convex bodies. Preprint. FA/0604595 32. : A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies. Funkts. Anal. Prilozh. 5(4), 28–37 (1971) (Russian) English translation in Funct. Anal. Appl. 5, 288–295 (1971) 33. : Dvoretzky’s theorem – thirty years later.

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

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