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We introduce a new formalism for defining and computing the volumes of completely general polytopes in any dimension. The expressions that we obtain for these volumes are independent of any triangulation, and manifestly depend only on the vertices of the underlying polytope. As one application of this formalism, we obtain new expressions for tree-level, n-point NMHV amplitudes in N=4 Super Yang-Mills (SYM) theory.

Volumes of polytopes without triangulations