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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**Extra info for Geometric Aspects of Functional Analysis: Israel Seminar 2001-2002**

**Example text**

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 , .