A simplex: an affinely independent vertex set

The polytope_region() whose vertices are affinely independent, so the hull is a simplex of dimension nrow(vertices) - 1.

Usage

simplex_region(vertices = NULL, .hv = NULL)

Arguments

vertices: (V, d) numeric matrix, one vertex per row.
.hv: Internal only: the underlying dual-representation record in both double and rational representation.

Value

A simplex_region, which is also a polytope_region().

Details

certify() additionally needs every vertex inside the standard simplex (the tetrahedron below fails this). Such simplices still chart, project and fit; they just cannot be certified.

Examples

# The 2-simplex in R^3, e.g. the entire multinomial parameter space:
simplex_region(vertices = diag(3))
#> <simplex_region>
#>   3 vertices in R^3
#>   facets: 3 inequalities, 1 equality
#>   vertices:
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1

# The plurality region `{theta : theta_1 <= theta_2}` within it:
simplex_region(vertices = rbind(c(0.5, 0.5, 0), c(0, 1, 0), c(0, 0, 1)))
#> <simplex_region>
#>   3 vertices in R^3
#>   facets: 3 inequalities, 1 equality
#>   vertices:
#>      [,1] [,2] [,3]
#> [1,]  0.5  0.5    0
#> [2,]  0.0  1.0    0
#> [3,]  0.0  0.0    1

# A tetrahedron in R^3, e.g. a piece of a triangulated Gaussian null:
simplex_region(
  vertices = rbind(c(0, 0, 0), c(1, 0, 0), c(0, 1, 0), c(0, 0, 1))
)
#> <simplex_region>
#>   4 vertices in R^3
#>   facets: 4 inequalities
#>   vertices:
#>      [,1] [,2] [,3]
#> [1,]    0    0    0
#> [2,]    1    0    0
#> [3,]    0    1    0
#> [4,]    0    0    1