frustum_of_simplex {volesti}

Compute the percentage of the volume of the simplex that is contained in the intersection of a half-space and the simplex.

Description

A half-space H is given as a pair of a vector a\in R^d and a scalar z0\in R s.t.: a^Tx\leq z0. This function calls the Ali's version of the Varsi formula to compute a frustum of the simplex.

Usage

frustum_of_simplex(a, z0)

Arguments

a

A d-dimensional vector that defines the direction of the hyperplane.

z0

The scalar that defines the half-space.

Value

The percentage of the volume of the simplex that is contained in the intersection of a given half-space and the simplex.

References

Varsi, Giulio, “The multidimensional content of the frustum of the simplex,” Pacific J. Math. 46, no. 1, 303–314, 1973.

Ali, Mir M., “Content of the frustum of a simplex,” Pacific J. Math. 48, no. 2, 313–322, 1973.

Examples

# compute the frustum of H: -x1+x2<=0
a=c(-1,1)
z0=0
frustum = frustum_of_simplex(a, z0)