Characterizing a valuated delta-matroid as a family of delta-matroids

Two characterizations are given for a valuated delta-matroid. Let (V,ℱ) be an even delta-matroid on a finite set V with the family ℱ of feasible sets. It is shown that a function δ : ℱ → ℝ is a valuation of (V,ℱ) if and only if, for each linear weighting p : V → ℝ, the maximizers of δ + p form the family of feasible sets of a delta-matroid. It is also shown that δ is a valuation if and only if its conjugate function is ‘locally bisubmodular ’ at each point.