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R be such that 2 i θi = 1. Denote According to Lemma 7, the random variable Y = n i=1 θi X i A Berry-Esseen type inequality for convex bodies with an unconditional basis sup |P (ε + Y ≥ t) − t∈R (t)| ≤ Cε2 , (59) with some universal constant C ≥ 1. The random variable Y has an even, log-concave density by Prékopa–Leindler. We may thus apply Lemma 9, and conclude from (59) that sup |P (α ≤ Y ≤ β) − [ (α) − α≤β (β)]| ≤ 2 sup |P (Y ≥ t) − t∈R (t)| ≤ C ε2 . The theorem is thus proven. Appendix: Proof of Theorem 2 With Cédric Villani’s permission, we reproduce below the proof of Theorem 2 from his book [40, Sect.

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