A quick estimate for the volume of a polyhedron

Let P be a bounded polyhedron defined as the intersection of the non-negative orthant ℝ ₊ ⁿ and an affine subspace of codimension m in ℝⁿ. We show that a simple and computationally efficient formula approximates the volume of P within a factor of γ^m, where γ > 0 is an absolute constant. The formula provides the best known estimate for the volume of transportation polytopes from a wide family.