Comparing the sets of volume polynomials and Lorentzian polynomials

Abstract Given n convex bodies in the Euclidean space Rᵈ, we can find their volume polynomial which is a homogeneous polynomial of degree d in n variables. We consider the set of homogeneous polynomials of degree d in n variables that can be represented as the volume polynomial of any such given convex bodies. This set is a subset of the set of Lorentzian polynomials. Using our knowledge of operations that preserve the Lorentzian property, we give a complete classification of the cases for (n, d) when the two sets are equal.