Georgia Institute of Technology College of Sciences

That the ball minimizes surface area among all sets of fixed volume, was known since antiquity; this is equivalent to the fact that the ball is the unique set which yields equality in the isoperimetric inequality. But the isoperimetric inequality is only a very special case of quadratic inequalities about mixed volumes of convex bodies, whose equality cases were unknown since the time of Minkowski. This talk is about these quadratic inequalities and their unusual equality cases which we resolved using degenerate diffusions on the sphere. No background in geometry will be assumed. Joint work with Ramon van Handel.