The Volume of a Crosspolytope Truncated by a Halfspace.

In this paper, we consider the computation of the volume of an n-dimensional crosspolytope truncated by a halfspace. Since a crosspolytope has exponentially many facets, we cannot efficiently compute the volume by dividing the truncated crosspolytope into simplices. We show an O(n(6)) time algorithm for the computation of the volume. This makes a contrast to the 0-1 knapsack polytope, whose volume is #P-hard to compute. The paper is interested in the computation of the volume of the truncated crosspolytope because we conjecture the following question may have an affirmative answer: Does the existence of a polynomial time algorithm for the computation of the volume of a polytope K imply the same for K's geometric dual? We give one example where the answer is yes.