A classical theorem of Minkowski asserts that the volume of a linear combination of convex bodies is a homogeneous polynomial with nonnegative coefficients. Since then, numerous inequalities among these coefficients have been discovered, with interesting applications in algebraic geometry and combinatorics. In this talk, I will discuss the inverse problem to this question: given a homogeneous polynomial, can one decide whether it arises as a volume polynomial? I will present known cases as well as new results based on joint work with Huang, Huh, Michałek, and Wang.
