CONTENTS INTRODUCTION vii 1. CONCENTRATION FUNCTIONS AND INEQUALITIES 1 1.1 First examples 1 1.2 Concentration functions 3 1.3 Deviation inequalities 5 1.4 Observable diameter 14 1.5 Expansion coefficient 15 1.6 Laplace bounds and infimum-convolutions 16 Notes and Remarks 21 2. ISOPERIMETRIC AND FUNCTIONAL EXAMPLES 23 2.1 Isoperimetric examples 23 2.2 Brunn-Minkowski inequalities 32 2.3 Semigroup tools 38 Notes and Remarks 44 3 . CONCENTRATION AND GEOMETRY 47 3.1 Spectrum and concentration 47 3.2 Spectral and diameter bounds 53 3.3 Levy families 55 3.4 Topological applications 57 3.5 Euclidean sections of convex bodies 60 Notes and Remarks 65 4. CONCENTRATION IN PRODUCT SPACES 67 4.1 Martingale methods 67 4.2 Convex hull approximation 72 4.3 Control by several points 79 4.4 Convex infimum-convolution 82 4.5 The exponential distribution 83 Notes and Remarks 89 5. ENTROPY AND CONCENTRATION 91 5.1 Logarithmic Sobolev inequalities and concentration 91 5.2 Product measures 97 5.3 Modified logarithmic Sobolev inequalities 101 5.4 Discrete settings 108
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