Seminars

• The Brascamp-Lieb inequalities (winter 2012)
  Barthe's proof of the forward and reverse Brascamp-Lieb inequalities.
• References for seminar on K-convexity (fall 2011)
• Covering numbers (winter 2010)
  Covering one set by translates of another. Covering from inside. Separating points by disjoint translates of a set. Volume estimates for covering numbers and separation numbers.
• Volume ratio (fall 2009, winter 2010)
  The volume ratio of a convex body. The simplex (in general) and the cube (among symmetric convex bodies) have the highest volume ratio (Ball). Most sections of a convex body are somewhat Euclidean; how close to Euclidean depends on the volume ratio (Kašin, Szarek, Tomczak-Jaegermann). High-dimensional convex bodies are spiky. Every convex body has two orthogonal somewhat Euclidean sections of half-dimension. The intersection of a symmetric convex body with a suitably rotated copy of itself is somewhat Euclidean. How to compute the volume ratio of the unit balls of the p-norms. How to use an epsilon net to control a Lipschitz function on the sphere. How to obtain an epsilon net of reasonable size. Comparison of somewhat Euclidean sections from volume ratio with nearly Euclidean sections from M(K) (as in Milman's proof of Dvoretzky's theorem). Converting between integrals on the sphere and integrals in Gaussian space. Of symmetric convex bodies in John's position, the cube has highest "Gaussian volume ratio"; as a corollary, among such bodies, the cube has the lowest value of M(K).