Graduate Student / Postdoc Seminar

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Minkowski Problems, Singular Monge-Ampere PDEs, and Measure Concentration

Speaker: Deane Yang, Courant Institute of Mathematical Sciences

Location: Warren Weaver Hall 1302

Date: Friday, March 8, 2019, 1 p.m.

Synopsis:

One hundred years ago, Minkowski studied the problem of constructing a convex polytope by prescribing the areas and outer unit normals of its faces. He identified the necessary and sufficient conditions for a set of positive numbers and unit vectors to be the areas and outer unit normals of a polytope's faces. The smooth analogue of this problem asks: Given a smooth positive function on the unit sphere, when is it the Gauss curvature (viewed as a function of the outer unit normal) of a smooth closed convex hypersurface? This question is equivalent to an elliptic Monge-Ampere equation on the sphere. Twenty years later, Aleksandrov and, independently, Fenchel and Jessen introduced concept of the surface area measure of a convex body and solved the most general version of the Minkowski problem: When is a finite Borel measure on the unit sphere the surface area measure of a convex body? In recent years Erwin Lutwak, Gaoyong Zhang, and I, building on earlier work of Erwin Lutwak, showed that the Minkowski problem is just one in a rich family of problems about prescribing geometric measures of convex bodies. These are equivalent to singular degenerate elliptic Monge-Ampere equations, where the inhomogeneous term is a measure instead of a function. In this talk I will describe recent progress made by Erwin Lutwak, Gaoyong Zhang, me, and our collaborators on when solutions exist for these problems. In particular, I will describe newly discovered geometric phenomena, involving measure concentration, which arise naturally as obstructions to solutions of these generalized Minkowski problems.