A quick estimate for the volume of a polyhedron

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