Volume of Line Bundles via Valuation Vectors (Different from Okounkov Bodies)

Up to a factor 1/n!, the volume of a big line bundle agrees with the Euclidean volume of its Okounkov body. The latter is the convex hull of top rank valuation vectors of sections, all with respect to a single flag. In this paper, we give a new volume formula, valid in the ample cone. It is also based on top rank valuation vectors, but mixes data coming from several different flags.