The diabolical bubbles of H. Minkowski

Ramon van Handel, Princeton University
Fine Hall 314

The isoperimetric theorem states that the ball minimizes surface area among all bodies of the same volume. For convex bodies, volume and surface area are two examples of a large family of natural geometric parameters called mixed volumes (i.e., coefficients of the volume polynomial). It is a long-standing question in convex geometry, dating back to a classical 1903 paper of Minkowski, what happens when one constrains other mixed volumes in the isoperimetric problem. The extremal bodies turn out to be strikingly bizarre:in particular, they can be non-smooth and non-unique.The question comes down to understanding the extremals in certain cases of the Alexandrov-Fenchel inequality, which has connections with several different areas of mathematics. In joint work with Yair Shenfeld, we were able to fully characterize these extremals in Minkowski's original setting. I will describe how these extremals come about from the construction and analysis of certain highly degenerate elliptic operators.