Generalized permutahedra form a rich class of polytopes that naturally appear in many areas such as combinatorics, optimization and statistics. We study functions on generalized permutahedra that behave linearly under dilation and taking Minkowski sums. We give a complete classification of all such functions that are additionally positive, translation-invariant and invariant under permutations of the coordinates: they form a simplicial cone and we explicitly describe the generators. We apply our results to Ehrhart polynomials of generalized permutahedra. This is joint work with Mohan Ravichandran.