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Klartag 26. : On a certain converse of Hölder’s inequality. Linear operators and approximation. Proc. , Oberwolfach, (1971). Int. Ser. Numer. , vol. 20, pp. 182–184. Birkhäuser, Basel (1972) 27. : A property of Logarithmic concave functions I+II. Indag. , New Ser. 15, 505–521 (1953) (also known as Nederl. Akad. Wetensch. Proc. Ser. A. 56) 28. : Almost spherical sections; their existence and their applications. Jahresber. Dtsch. -Ver. 39–61 (1992) 29. : The geometry of logconcave functions and sampling algorithms.

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.

G(x) + f(x)]dx ≤ Cn (20) |g(x) − f(x)|dx ≥ 1 − T g = 1, then according to (18) and (19), |g(x) − f(x)|dx ≤ Rn \T The lemma follows by adding inequalities (17) and (20). 122 B. 1 allows us to convolve our log-concave function with a small gaussian. 1. We sketch the main points of difference between the proofs. 5. Let n ≥ 2 be an integer, let α ≥ 10, and let f : Rn → [0, ∞) be an isotropic, log-concave function. Assume that sup Mf (θ, t) ≤ e−5αn log n + inf Mf (θ, t) for all t ∈ R. (21) θ∈S n−1 θ∈S n−1 Denote g = f ∗ γn,n−α , where ∗ stands for convolution.

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