Polytope volume computation

Lawrence, Jim

doi:10.1090/S0025-5718-1991-1079024-2

Polytope volume computation
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Math. Comp. 57 (1991), 259-271 Request permission

Abstract:

A combinatorial form of Gram’s relation for convex polytopes can be adapted for use in computing polytope volume. We present an algorithm for volume computation based on this observation. This algorithm is useful in finding the volume of a polytope given as the solution set of a system of linear inequalities, $P = \{ x \in {\mathbb {R}^n}:Ax \leq b\}$ . As an illustration we compute a formula for the volume of a projective image of the n-cube. From this formula we deduce that, when A and b have rational entries (so that the volume of P is also a rational number), the number of binary digits in the denominator of the volume cannot be bounded by a polynomial in the total number of digits in the numerators and denominators of entries of A and b . This settles a question posed by Dyer and Frieze.

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