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© 2015

Approaching the Kannan-Lovász-Simonovits and Variance Conjectures

Benefits

  • Provides a rapid introduction to Asymptotic Geometric Analysis

  • Presents a modern approach to two central problems in Asymptotic Geometric Analysis: The Kannan-Lovasz-Simmonovits Conjecture and the Variance (or Thin Shell) Conjecture

  • Details the results and methods in as elementary a way as possible, aiming the presentation at a wide audience

  • Introduces interested readers to a fascinating area of research

  • Provides a useful starting point for young researchers who wish to work on these conjectures

Book

Part of the Lecture Notes in Mathematics book series (LNM, volume 2131)

Table of contents

  1. Front Matter
    Pages i-x
  2. David Alonso-Gutiérrez, Jesús Bastero
    Pages 1-64
  3. David Alonso-Gutiérrez, Jesús Bastero
    Pages 65-101
  4. David Alonso-Gutiérrez, Jesús Bastero
    Pages 103-135
  5. Back Matter
    Pages 137-150

About this book

Introduction

Focusing on two central conjectures from the field of Asymptotic Geometric Analysis, the Kannan-Lovász-Simonovits spectral gap conjecture and the variance conjecture, these Lecture Notes present the theory in an accessible way, so that interested readers, even those who are not experts in the field, will be able to appreciate the topics treated. Employing a style suitable for professionals with little background in analysis, geometry or probability, the work goes directly to the connection between isoperimetric-type inequalities and functional inequalities, allowing readers to quickly access the core of these conjectures.

In addition, four recent and important results concerning this theory are presented. The first two are theorems attributed to Eldan-Klartag and Ball-Nguyen, which relate the variance and the KLS conjectures, respectively, to the hyperplane conjecture. The remaining two present in detail the main ideas needed to prove the best known estimate for the thin-shell width given by Guédon-Milman, and an approach to Eldan’s work on the connection between the thin-shell width and the KLS conjecture.

Keywords

46Bxx,52Axx,60-XX,28Axx. Convex bodies Isoperimetric inequalities Poincaré's inequalities for log-concave probabilities Spectral gap estimates Variance conjecture

Authors and affiliations

  1. 1.Departament de Matemàtiques and IMACUniversitat Jaume ICastelló de la PlanaSpain
  2. 2.Departamento de Matemáticas and IMACUniversidad de ZaragozaZaragozaSpain

Bibliographic information

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Reviews

“This book serves as an excellent and well-written introduction to a fascinating and active research subject. It is a must have for specialists as well as students interested in diving into this subject, but it is also suitable for mathematicians with a different focus who are interested in a taste of this theory. It can also easily be used as a basis for an advanced course.” (Ronen Eldan, Mathematical Reviews, October, 2015)