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.
The exact volume of the input polytope, for zonotopes, simplices in V-representation and polytopes with known exact volume
E. Gover and N. Krikorian, “Determinants and the Volumes of Parallelotopes and Zonotopes,” Linear Algebra and its Applications, 433(1), 28 - 40, 2010.
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# 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) # 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) # compute the exact volume the 10-dimensional cross polytope P = gen_cross(10,'V') vol = exact_vol(P)
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