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.