Definition

1. for each
2. for each

Properties

1. is nonempty if and only if and that is nonempty if and only if
2. Given any extended polymatroid there is a unique submodular function such that and
3. If r is integral, 1-Lipschitz
   Lipschitz continuity
   In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...
   
   , and then r is the rank-function of a matroid, and the polymatroid is the independent set polytope, so-called since Edmonds showed it is the convex hull
   Convex hull
   In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
   
    of the characteristic vectors of all independent sets of the matroid.
4. For a supermodular
   Supermodular
   In mathematics, a functionf\colon R^k \to Ris supermodular iff + f \geq f + ffor all x, y \isin Rk, where x \vee y denotes the componentwise maximum and x \wedge y the componentwise minimum of x and y.If −f is supermodular then f is called submodular, and if the inequality is changed to an...
   
    f one analogously may define the contrapolymatroid

This analogously generalizes the dominant of the spanning set polytope

Polytope

In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...

of matroids.