Analytical and Probabilistic Methods in the Geometry of Convex Bodies

Title page
Concentration inequalities and geometry of convex-bodies / Olivier Guédon, Piotr Nayar and Tomasz Tkocz
Analytic methods in convex geometry / Dmitry Ryabogin and Artem Zvavitch
Lecture I: Concentration inequalities and geometry of convex-bodies
1. Introduction
2. Brascamp-Lieb inequalities in a geometric context
2.1. Motivation and formulation of the inequality
2.2. The proof
2.3. Consequences of the Brascamp-Lieb inequality
2.4. Notes and comments
3. Borell and Prékopa-Leindler type inequalities. Ball's bodies
3.1. Brunn-Minkowski inequality
3.2. Functional version of the Brunn-Minkowski inequality
3.3. Functional version of the Blaschke-Santalo inequality
3.4. Borell and Ball functional inequalities
3.5. Consequences in convex geometry
3.6. Notes and comments
4. Concentration of measure. Dvoretzky's Theorem
4.1. Isoperimetric problem
4.2. Concentration inequalities
4.3. Dvoretzky's Theorem
4.4. Comparison of moments of a norm of a Gaussian vector
4.5. Notes and comments
5. Reverse Hölder inequalities and volumes of sections of convex bodies
5.1. Berwald's inequality and its extensions
5.2. Some concentration inequalities
5.3. Kahane-Khinchin type inequalities
5.4. Notes and comments
6. Concentration of mass of a log-concave measure
6.1. The result
6.2. Z_p-bodies associated with a measure
6.3. The final step
6.4. Notes, comments and further reading
References
Lecture II: Analytic methods in convex geometry
1. General plan
2. Short introduction
2.1. Main definitions and facts
2.2. Steiner symmetrlzatlon
2.3. Brunn-Minkowski inequality
3. Duality and volume, the first look
3.1. R² case
3.2. Unconditional case
3.3. Local minimum
4. Zonoids and zonotopes
4.1. Mahler conjecture: case of zonoids
5. Radon and Cosine transforms
6. Local and equatorial characterizations of zonoids
6.1. A motivating example
6.2. Local equatorial characterization
6.3. Local characterization
7. What information can uniquely determine a convex body?
7.1. Questions of Bonnensen and Klee
8. Isomorphic version of the Mahler conjecture
9. Part I. Main Idea: application of the Paley-Wiener Theorem
10. Part II. Adjustment of the main idea to the Bergman space A²(T_K)
10.1. Rothaus-Koranyi-Hsin formula for the reproducing kernel in T_K
10.2. Estimates related to the Bergman kernel in the tube domain
11. Part III. Auxiliary construction
11.1. Construction of the plurisubharmonic φ on T_K
11.2. Construction of the Mexican-hat function g supported on K_C
12. Part IV. Construction of the analytic function F
12.1. Almost exact lower bound on etc
12.2. Exact lower bound on ... . The tensor power trick
Appendices A. Paley-Wiener Theorem
B. Bergman spaces
B.l. Reproducing kernel
C. Hörmander Theorem and some of its applications
C.l. Convexity and pseudo-convexity
C.2. Formulation of the Hörmander Theorem
C.3. From subharmonic to analytic. Examples in the case n = 1
C.4. Non-trivial F == F(φ,g) obtained by applying the Hörmander Theorem in T_K
D. Proof of Hörmander's Theorem
References