Caltech/UCLA Joint Analysis Seminar
It is a basic fact of convexity that the volume of convex bodies is a polynomial, whose coefficients (mixed volumes) define a large family of natural geometric parameters. A fundamental result of convex geometry, the Alexandrov-Fenchel inequality, states that these coefficients are log-concave. This result proves to have striking connections with other areas of mathematics, such as combinatorics and algebraic geometry.
There is a long-standing problem surrounding the Alexandrov-Fenchel inequality that has remained open since the original works of Minkowski (1903) and Alexandrov (1937): in what cases is equality attained? This question corresponds to the solution of certain unusual isoperimetric problems, whose extremal bodies turn out to be numerous and strikingly bizarre. With Y. Shenfeld, we recently succeeded to settle this problem completely in the setting of convex polytopes, as well as to develop new tools for the study of general convex bodies. In this talk, I aim to sketch what the extremals look like and to indicate some combinatorial, analytic, and geometric issues that arise in their characterization.