Titlebook: Geometric Aspects of Functional Analysis; Israel Seminar (GAFA Bo‘a(chǎn)z Klartag,Emanuel Milman Book 2020 Springer Nature Switzerland AG 2020 A

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,The Lower Bound for Koldobsky’s Slicing Inequality via Random Rounding,and all . . Our bound is optimal, up to the value of the universal constant. It improves slightly upon the results of the first named author and Koldobsky, which included a doubly-logarithmic error. The proof is based on an efficient way of discretizing the unit sphere.

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Two-Sided Estimates for Order Statistics of Log-Concave Random Vectors,uncorrelated coordinates. Our bounds are exact up to multiplicative universal constants in the unconditional case for all . and in the isotropic case for .?≤?.???... We also derive two-sided estimates for expectations of sums of . largest moduli of coordinates for some classes of random vectors.

ABHOR

,Further Investigations of Rényi Entropy Power Inequalities and an Entropic Characterization of s-CoBobkov and Chistyakov (IEEE Trans Inform Theory 61(2):708–714, 2015) fails when the Rényi parameter .?∈?(0, 1), we show that random vectors with .-concave densities do satisfy such a Rényi entropy power inequality. Along the way, we establish the convergence in the Central Limit Theorem for Rényi en

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Concentration of the Intrinsic Volumes of a Convex Body,quence of intrinsic volumes. The main result states that the intrinsic volume sequence concentrates sharply around a specific index, called the central intrinsic volume. Furthermore, among all convex bodies whose central intrinsic volume is fixed, an appropriately scaled cube has the intrinsic volum

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Two Remarks on Generalized Entropy Power Inequalities,nicity and entropy comparison of weighted sums of independent identically distributed log-concave random variables. We also present a complex analogue of a recent dependent entropy power inequality of Hao and Jog, and give a very simple proof.

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On the Geometry of Random Polytopes,ric random variable that has variance 1, let Γ?=?(..) be an .?×?. random matrix whose entries are independent copies of ., and set .., …, .. to be the rows of Γ. Then under minimal assumptions on . and as long as .?≥?.., with high probability

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