SliceOfSimplex {volesti}R Documentation

Compute the percentage of the volume of the unit simplex that is contained in the intersection of a half-space and the unit 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≤q z0. This function calls the Ali's version of the Varsi formula to compute a frustum of the unit simplex.

Usage

SliceOfSimplex(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 unit simplex that is contained in the intersection of a given half-space and the unit 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 = SliceOfSimplex(a, z0)

[Package volesti version 1.0.3 Index]