Download A Bernstein Property of Affine Maximal Hypersurfaces by Klartag B. PDF

Download A Bernstein Property of Affine Maximal Hypersurfaces by Klartag B. PDF

By Klartag B.

Show description

Read or Download A Bernstein Property of Affine Maximal Hypersurfaces PDF

Best nonfiction_1 books

Beware of Cat: And Other Encounters of a Letter Carrier

One sunny day on his postal path, Vincent Wyckoff crosses the trail of an aged gentleman whistling for his misplaced parakeet. The previous guy is disenchanted, and Wyckoff strikes down the block slowly, taking a look low and high, hoping to identify the little chicken. He reaches the man’s apartment and provides sympathy to his spouse, who smiles unfortunately and says, “We haven’t had that chicken for twenty-five years.

Scientific American (May 2002)

Journal is in first-class situation. m105

Fine Wood Working. Checkered Bowls Winter

Woodworking-making checkered bowls and lots more and plenty extra.

Extra resources for A Bernstein Property of Affine Maximal Hypersurfaces

Example text

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 .

Download PDF sample

Rated 4.68 of 5 – based on 5 votes
Comments are closed.