Geometric Aspects of Functional Analysis: Israel Seminar by Francesco Bullo, Andrew D. Lewis

By Francesco Bullo, Andrew D. Lewis

The lawsuits of the Israeli GAFA seminar on Geometric point of useful research throughout the years 2001-2002 stick to the lengthy culture of the former volumes. They proceed to mirror the overall traits of the idea. a number of papers care for the cutting challenge and its family. a few care for the focus phenomenon and comparable issues. in lots of of the papers there's a deep interaction among chance and Convexity. the amount includes additionally a profound examine on approximating convex units through randomly selected polytopes and its relation to floating our bodies, a massive topic in Classical Convexity thought. all of the papers of this assortment are unique examine papers.

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Let us now turn to the second task: approximation of the µn -typical F by more canonical distributions. Namely, denote by G the distribution function √ where ζ is a standard normal random variable of the random variable ζ |X| n independent of the Euclidean norm |X| = (X12 + . . + Xn2 )1/2 . Clearly, G represents a mixture of a family of centered Gaussian measures on the line and has characteristic function 2 g(t) = Ee−t while F has characteristic function |X|2 /(2n) , t ∈ R, (8) Concentration of Distributions of Weighted Sums n f (t) = E cos k=1 tXk √ n .

Assume E|X|2 ≤ n. For all α > 0 and |t| ≤ |f (t) − g(t)| ≤ max{|X1 |, . . , |Xn |} √ >α . n 1 2 4 α t + 2P 9 Proof. By Taylor’s expansion, in the interval |s| ≤ 2 − s2 −u(s) e with u satisfying 0 ≤ u(s) ≤ for all k ≤ n, and α|t| ≤ 12 , n tXk √ n cos k=1 with 0 ≤ n k=1 √k) ≤ u( tX n − e t2 |X|2 2n = exp − 4 s 9 n ≥ cos k=1 1 2, we have cos(s) = Xk . Therefore, provided that | √ | ≤ α, n t2 |X|2 − 2n n tXk √ n ≥ e− tXk √ n u k=1 n tX √k 2 k=1 | n | √ k |2 maxk | tX n 1 9 1 2α , t2 |X|2 2n ≤ 2 α2 t4 |X| 9 n .

Xn,n ) according to formulas (9) and (8), respectively. Also, according to (3), denote by F (n) the corresponding average distribution functions. ,|Xn,n |} √ P{ > αn } → 0, as n → ∞. Then, by Lemma 4, for all n t ∈ R, |fn (t) − gn (t)| → 0, as n → ∞. On the other hand, the condition b) 2 2 readily implies gn (t) → e−t /2 , so fn (t) → e−t /2 . Thus, L(F (n) , Φ) → 0. Concentration of Distributions of Weighted Sums 35 Now, given an infinite sequence ε ∈ {−1, 1}∞ , denote by Tn (ε) its projection (ε1 , .

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