Home > Asymptotic Geometric Analysis: Concentration of Measure, Dvoretzky's Theorem, Milman's Reverse Brunn-Minkowski Inequality(English)
Asymptotic Geometric Analysis: Concentration of Measure, Dvoretzky's Theorem, Milman's Reverse Brunn-Minkowski Inequality(English)

Asymptotic Geometric Analysis: Concentration of Measure, Dvoretzky's Theorem, Milman's Reverse Brunn-Minkowski Inequality(English)

          
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About the Book

Chapters: Concentration of Measure, Dvoretzky's Theorem, Milman's Reverse Brunn-minkowski Inequality, Quotient of Subspace Theorem. Source: Wikipedia. Pages: 20. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics, concentration of measure (about a median) is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space theory. Informally, it states that Lipschitz functions that depend on many parameters are almost constant. The c.o.m. phenomenon was put forth in the early 1970-s by Vitali Milman in his works on the local theory of Banach spaces, extending an idea going back to the work of Paul Levy. It was further developed in the works of Milman and Gromov, Maurey, Pisier, Schechtman, Talagrand, Ledoux, and others. Let be a metric measure space, . Let where is the -extension of a set . The function is called the concentration rate of the space . The following equivalent definition has many applications: where the supremum is over all 1-Lipschitz functions, and the median (or Levy mean) is defined by the inequalities Informally, the space exhibits a concentration phenomenon if decays very fast as grows. More formally, a family of metric measure spaces is called a Levy family if the corresponding concentration rates satisfy and a normal Levy family if for some constants . For examples see below. The first example goes back to Paul Levy. According to the spherical isoperimetric inequality, among all subsets of the sphere with prescribed spherical measure, the spherical cap has the smallest -extension (for any ). Applying this to sets of measure (where ), one can deduce the following concentration inequality: , where are universal constants. Therefore form a normal Levy family. Vitali Milman applied this f...More: http: //booksllc.net/?id=788497


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Product Details
  • ISBN-13: 9781158336203
  • Publisher: Books LLC
  • Publisher Imprint: Books LLC
  • Height: 152 mm
  • No of Pages: 22
  • Spine Width: 1 mm
  • Weight: 45 gr
  • ISBN-10: 1158336209
  • Publisher Date: 15 Sep 2010
  • Binding: Paperback
  • Language: English
  • Series Title: English
  • Sub Title: Concentration of Measure, Dvoretzky's Theorem, Milman's Reverse Brunn-Minkowski Inequality
  • Width: 229 mm


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