Empty set (EmptySet)

EmptySet{N} <: ConvexSet{N}

Type that represents the empty set, i.e., the set with no elements.

∅

Alias for EmptySet{Float64}.

an_element(∅::EmptySet)

Return some element of an empty set.

Input

• ∅ – empty set

Output

An error.

area(∅::EmptySet)

Return the area of an empty set.

Input

• ∅ – empty set

Output

$0$.

chebyshev_center_radius(∅::EmptySet; [kwargs]...)

Compute the Chebyshev center and the corresponding radius of an empty set.

Input

• ∅ – empty set
• kwargs – further keyword arguments (ignored)

Output

An error.

complement(∅::EmptySet{N}) where {N}

Return the complement of an empty set.

Input

• ∅ – empty set

Output

The universe of the same dimension.

complement(U::Universe{N}) where {N}

Return the complement of an universe.

Input

• ∅ – universe

Output

The empty set of the same dimension.

diameter(∅::EmptySet, [p]::Real=Inf)

Return the diameter of an empty set. It is the maximum distance between any two elements of the set or, equivalently, the diameter of the enclosing ball (of the given $p$-norm) of minimal volume with the same center.

Input

• ∅ – empty set
• p – (optional, default: Inf) norm

Output

An error.

dim(∅::EmptySet)

Return the dimension of an empty set.

Input

• ∅ – an empty set

Output

The dimension of the empty set.

high(∅::EmptySet)

Return a vector with the highest coordinates of an empty set in each canonical direction.

Input

• ∅ – empty set

Output

An error.

Notes

See also high(∅::EmptySet, i::Int).

high(∅::EmptySet, i::Int)

Return the highest coordinate of an empty set in the given direction.

Input

• ∅ – empty set
• i – dimension of interest

Output

An error.

isbounded(∅::EmptySet)

Check whether an empty set is bounded.

Input

• ∅ – empty set

Output

true.

isempty(∅::EmptySet)

Check if the empty set is empty.

Input

• ∅ – empty set

Output

true.

isuniversal(∅::EmptySet{N}, [witness]::Bool=false) where {N}

Check whether an empty set is universal.

Input

• ∅ – empty set
• witness – (optional, default: false) compute a witness if activated

Output

• If witness option is deactivated: false
• If witness option is activated: (false, v) where $v ∉ S$

low(∅::EmptySet)

Return a vector with the lowest coordinates of an empty set in each canonical direction.

Input

• ∅ – empty set

Output

An error.

Notes

See also low(∅::EmptySet, i::Int).

low(∅::EmptySet, i::Int)

Return the lowest coordinate of an empty set in the given direction.

Input

• ∅ – empty set
• i – dimension of interest

Output

An error.

norm(∅::EmptySet, [p]::Real=Inf)

Return the norm of an empty set. It is the norm of the enclosing ball (of the given $p$-norm) of minimal volume that is centered in the origin.

Input

• ∅ – empty set
• p – (optional, default: Inf) norm

Output

An error.

radius(∅::EmptySet, [p]::Real=Inf)

Return the radius of an empty set. It is the radius of the enclosing ball (of the given $p$-norm) of minimal volume with the same center.

Input

• ∅ – empty set
• p – (optional, default: Inf) norm

Output

An error.

rand(::Type{EmptySet}; [N]::Type{<:Real}=Float64, [dim]::Int=2,
     [rng]::AbstractRNG=GLOBAL_RNG, [seed]::Union{Int, Nothing}=nothing)

Create an empty set (note that there is nothing to randomize).

Input

• EmptySet – type for dispatch
• N – (optional, default: Float64) numeric type
• dim – (optional, default: 2) dimension
• rng – (optional, default: GLOBAL_RNG) random number generator
• seed – (optional, default: nothing) seed for reseeding

Output

The (only) empty set of the given numeric type and dimension.

rectify(∅::EmptySet)

Concrete rectification of an empty set.

Input

• ∅ – empty set

Output

The empty set.

reflect(∅::EmptySet)

Concrete reflection of an empty set.

Input

• ∅ – empty set

Output

The same empty set.

vertices_list(∅::EmptySet)

Return the list of vertices of an empty set.

Input

• ∅ – empty set

Output

The empty list of vertices, as the empty set does not contain any vertices.

vertices(∅::EmptySet)

Construct an iterator over the vertices of an empty set.

Input

• ∅ – empty set

Output

The empty iterator, as the empty set does not contain any vertices.

volume(∅::EmptySet{N}) where {N}

Return the volume of an empty set.

Input

• ∅ – empty set

Output

$0$.

∈(x::AbstractVector, ∅::EmptySet)

Check whether a given point is contained in an empty set.

Input

• x – point/vector
• ∅ – empty set

Output

false.

Examples

julia> [1.0, 0.0] ∈ ∅(2)
false

linear_map(M::AbstractMatrix{N}, ∅::EmptySet{N}) where {N}

Return the linear map of an empty set.

Input

• M – matrix
• ∅ – empty set

Output

An empty set.

ρ(d::AbstractVector, ∅::EmptySet)

Evaluate the support function of an empty set in a given direction.

Input

• d – direction
• ∅ – empty set

Output

An error.

σ(d::AbstractVector, ∅::EmptySet)

Return the support vector of an empty set.

Input

• d – direction
• ∅ – empty set

Output

An error.

translate(∅::EmptySet, v::AbstractVector)

Translate (i.e., shift) an empty set by a given vector.

Input

• ∅ – empty set
• v – translation vector

Output

The empty set.

plot_recipe(∅::EmptySet{N}, [ε]=zero(N)) where {N}

Convert an empty set to a sequence of points for plotting. In the special case of an empty set, the sequence is empty.

Input

• ∅ – empty set
• ε – (optional, default: 0) ignored, used for dispatch

Output

An empty array.