By Klartag B.
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Extra resources for A Bernstein Property of Affine Maximal Hypersurfaces
By Taylor’s theorem, there exists a constant R = R(ϕ) with ϕ(y) − ϕ(x) ≤ |∇ϕ(x)| · |x − y| + R|x − y|2 ∀x, y ∈ Rn . (61) We may assume that sup |h| > 0 (otherwise, the theorem holds trivially), and let ε > 0 be smaller than 1/ sup |h|. Then µε is a non-negative measure on Rn . Let γ be any coupling of µ and µε . We see that hϕdµ = Rn 1 ε ϕd [µε − µ] = Rn 1 ε [ϕ(y) − ϕ(x)] dγ (x, y). R n ×R n γ 2 Write W2 (µ, µε ) = Rn ×Rn |x − y| dγ (x, y). According to (61) and to the Cauchy-Schwartz inequality, hϕdµ ≤ Rn ≤ 1 ε 1 ε |∇ϕ(x)| · |x − y|dγ (x, y) + R n ×R n γ |∇ϕ(x)|2 dµ(x) · W2 (µ, µε ) + Rn R ε |x − y|2 dγ (x, y) R n ×R n R γ W (µ, µε )2 .
Proof of Theorem 1 Let θ1 , . . , θn ε = 10 satisfies 123 n 4 i=1 θi . ∈ 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 .
60) Since µ is compactly supported, it is enough to restrict attention to compactly supported functions ϕ. Fix such a test function ϕ. Then the second derivatives of ϕ are bounded on Rn . By Taylor’s theorem, there exists a constant R = R(ϕ) with ϕ(y) − ϕ(x) ≤ |∇ϕ(x)| · |x − y| + R|x − y|2 ∀x, y ∈ Rn . (61) We may assume that sup |h| > 0 (otherwise, the theorem holds trivially), and let ε > 0 be smaller than 1/ sup |h|. Then µε is a non-negative measure on Rn . Let γ be any coupling of µ and µε .