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volesti (version 1.1.2)

exact_vol: Compute the exact volume of (a) a zonotope (b) an arbitrary simplex in V-representation or (c) if the volume is known and declared by the input object.

Description

Given a zonotope (as an object of class Zonotope), this function computes the sum of the absolute values of the determinants of all the \(d \times d\) submatrices of the \(m\times d\) matrix \(G\) that contains row-wise the \(m\) \(d\)-dimensional segments that define the zonotope. For an arbitrary simplex that is given in V-representation this function computes the absolute value of the determinant formed by the simplex's points assuming it is shifted to the origin.

Usage

exact_vol(P)

Arguments

P

A polytope

Value

The exact volume of the input polytope, for zonotopes, simplices in V-representation and polytopes with known exact volume

References

E. Gover and N. Krikorian, “Determinants and the Volumes of Parallelotopes and Zonotopes,” Linear Algebra and its Applications, 433(1), 28 - 40, 2010.

Examples

Run this code
# NOT RUN {
# compute the exact volume of a 5-dimensional zonotope defined by the Minkowski sum of 10 segments
Z = gen_rand_zonotope(2, 5)
vol = exact_vol(Z)

# }
# NOT RUN {
# compute the exact volume of a 2-d arbitrary simplex
V = matrix(c(2,3,-1,7,0,0),ncol = 2, nrow = 3, byrow = TRUE)
P = Vpolytope(V = V)
vol = exact_vol(P)
# }
# NOT RUN {
# compute the exact volume the 10-dimensional cross polytope
P = gen_cross(10,'V')
vol = exact_vol(P)
# }

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