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The isoperimetric theorem states that the ball minimizes surface areaÌýamong all bodies of the same volume. For convex bodies, volume and surfaceÌýarea are two examples of a large family of natural geometric parametersÌýcalled mixed volumes (i.e., coefficients of the volume polynomial). It isÌýa long-standing question in convex geometry, dating back to a classicalÌý1903 paper of Minkowski, what happens when one constrains other mixedÌývolumes in the isoperimetric problem. The extremal bodies turn out to beÌýstrikingly bizarre:in particular, they can be non-smooth and non-unique.The question comes down to understanding the extremals in certain cases ofÌýthe Alexandrov-Fenchel inequality, which has connections with severalÌýdifferent areas of mathematics. In joint work with Yair Shenfeld, we wereÌýable to fully characterize these extremals in Minkowski's originalÌýsetting. I will describe how these extremals come about from theÌýconstruction and analysis of certain highly degenerate elliptic operators.

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