volesti (version 1.1.2-9)

frustum_of_simplex: 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)

Value

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

Arguments

a

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

z0

The scalar that defines the half-space.

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)