In mathematics, a polymatroid is a polytope associated with a submodular function. The notion was introduced by Jack Edmonds in 1970. It is also described as the multiset analogue of the matroid. Definition Let be a finite set and a non-decreasing submodular function, that is, for each we have , and for each we have . We define the polymatroid associated to to be the following polytope: . When we allow the entries of to be negative we denote this polytope by , and call it the extended polymatroid associated to . An equivalent definition Let be a finite set and . We call the modulus of to be the sum of all of its entries, denoted , and denote whenever for every (notice that this gives an order to ). A polymatroid on the ground set is a nonempty compact subset in , the set of independent vectors, such that: We have that if , then for every : If with , then there is a vector such that . This definition is equivalent to the one described before, where is the function defined by for every . Relation to matroids To every matroid on the ground set we can associate the set , where for every we have that By taking the convex hull of we get a polymatroid, in the sense of the second definition, associated to the rank function of . Relation to generalized permutahedra Because generalized permutahedra can be constructed from submodular functions, and every generalized permutahedron has an associated submodular function, we have that there should be a correspondence between generalized permutahedra and polymatroids. In fact every polymatroid is a generalized permutahedron that has been translated to have a vertex in the origin. This result suggests that the combinatorial information of polymatroids is shared with generalized permutahedra. Properties is nonempty if and only if and that is nonempty if and only if . Given any extended polymatroid there is a unique submodular function such that and . Contrapolymatroids For a supermodular f one analogously may define the contrapolymatroid This analogously generalizes the dominant of the spanning set polytope of matroids. Discrete polymatroids When we only focus on the lattice points of our polymatroids we get what is called, discrete polymatroids. Formally speaking, the definition of a discrete polymatroid goes exactly as the one for polymatroids except for where the vectors will live in, instead