Contents

Preface ix

Chapter 1. Background from asymptotic convex geometry 1

1.1. Convex bodies 1

1.2. Brunn–Minkowski inequality 4

1.3. Applications of the Brunn-Minkowski inequality 8

1.4. Mixed volumes 12

1.5. Classical positions of convex bodies 16

1.6. Brascamp-Lieb inequality and its reverse form 22

1.7. Concentration of measure 25

1.8. Entropy estimates 34

1.9. Gaussian and sub-Gaussian processes 38

1.10. Dvoretzky type theorems 43

1.11. The -position and Pisier’s inequality 50

1.12. Milman’s low M

∗-estimate

and the quotient of subspace theorem 52

1.13. Bourgain-Milman inequality and the M-position 55

1.14. Notes and references 58

Chapter 2. Isotropic log-concave measures 63

2.1. Log-concave probability measures 63

2.2. Inequalities for log-concave functions 66

2.3. Isotropic log-concave measures 72

2.4. ψα-estimates 78

2.5. Convex bodies associated with log-concave functions 84

2.6. Further reading 94

2.7. Notes and references 100

Chapter 3. Hyperplane conjecture and Bourgain’s upper bound 103

3.1. Hyperplane conjecture 104

3.2. Geometry of isotropic convex bodies 108

3.3. Bourgain’s upper bound for the isotropic constant 116

3.4. The ψ2-case 123

3.5. Further reading 128

3.6. Notes and references 134

Chapter 4. Partial answers 139

4.1. Unconditional convex bodies 139

4.2. Classes with uniformly bounded isotropic constant 144

4.3. The isotropic constant of Schatten classes 150

4.4. Bodies with few vertices or few facets 155

v