# The Volume of a Crosspolytope Truncated by a Halfspace

## Abstract

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.

## Keywords

Polynomial time algorithm Volume computation Geometric duality## References

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