Differential Geometry

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By Christian Houdre, Michel Ledoux, Emanuel Milman, Mario Milman

The quantity includes the court cases of the overseas workshop on focus, practical Inequalities and Isoperimetry, held at Florida Atlantic collage in Boca Raton, Florida, from October 29-November 1, 2009.

The interactions among focus, isoperimetry and sensible inequalities have ended in many major advances in sensible research and chance theory.

vital development has additionally taken position in combinatorics, geometry, harmonic research and mathematical physics, to call yet a number of fields, with fresh new purposes in random matrices and data idea. This ebook should still entice graduate scholars and researchers attracted to the attention-grabbing interaction among research, chance, and geometry.

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Additional info for Concentration, Functional Inequalities and Isoperimetry: International Workshop on Concentration, Functional Inequalities and Isoperiometry, October ... Boca Ra

Example text

N = 1. At this step we involve the following observation made by D. Cordero-Erausquin, M. Fradelizi and B. Maurey in their study and proof of the so-called B-conjecture, cf. [CE-F-M], Theorem 1. It is stated below as a lemma, where D(λ) is treated as a linear map. 2 (D. Cordero-Erausquin, M. Fradelizi and B. Maurey [CE-F-M]). For any symmetric convex body K in Rn , the function (t1 , . . , tn ) −→ γ(D(et1 , . . , etn )(K)) is log-concave on Rn . 1, introduce the function on Rn−1 v(t1 , . .

1), and the (Shannon) entropy power of X is N (X) = e2h(X)/n . We limit ourselves to random vectors X with h(X) < +∞; in this case, N (X) is a non-negative real number. Building on work of [32] and resolving a conjecture they made, [33] recently showed the following result. 1. Let X1 , . . , XM be independent Rn -valued random vectors, such that the entropy of each exists and is finite. Let β be a fractional partition using a collection G of subsets of [M ]. Then N (X1 + . . + XM ) ≥ βs N s∈G Xj .

Some applications of duality relations. , 1469, Springer, Berlin, 1991. [M-P] Milman, V. , and Pajor, A. Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. , 1376, Springer, Berlin, 1989. [P] Pisier, G. The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics, 94. Cambridge University Press, Cambridge, 1989, xvi+250 pp. C. The difference body of a convex body. Arch. Math. (Basel) 8 (1957), 220–233. C. Convex bodies associated with a given convex body.

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