Empty set (EmptySet)

EmptySet{N<:Real} <: LazySet{N}

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

∅

An EmptySet instance of type Float64.

dim(∅::EmptySet)

Return the dimension of the empty set, which is -1 by convention.

Input

• ∅ – an empty set

Output

-1 by convention.

σ(d::AbstractVector{N}, ∅::EmptySet{N}) where {N<:Real}

Return the support vector of an empty set.

Input

• d – direction
• ∅ – empty set

Output

An error.

ρ(d::AbstractVector{N}, ∅::EmptySet{N}) where {N<:Real}

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

Input

• d – direction
• ∅ – empty set

Output

An error.

∈(x::AbstractVector{N}, ∅::EmptySet{N}) where {N<:Real}

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

Input

• x – point/vector
• ∅ – empty set

Output

The output is always false.

Examples

julia> [1.0, 0.0] ∈ ∅
false

an_element(∅::EmptySet)

Return some element of an empty set.

Input

• ∅ – empty set

Output

An error.

rand(::Type{EmptySet}; [N]::Type{<:Real}=Float64, [dim]::Int=0,
     [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: 0) dimension (is ignored)
• 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.

isbounded(∅::EmptySet)

Determine whether an empty set is bounded.

Input

• ∅ – empty set

Output

true.

isempty(∅::EmptySet)

Return if the empty set is empty or not.

Input

• ∅ – empty set

Output

true.

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

Check whether an empty 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 $v ∉ S$, although we currently throw an error

norm(S::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

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

Output

An error.

radius(S::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

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

Output

An error.

diameter(S::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

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

Output

An error.

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

Return the linear map of an empty set.

Input

• M – matrix
• ∅ – empty set

Output

The empty set.

translate(∅::EmptySet{N}, v::AbstractVector{N}) where {N<:Real}

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]::N=zero(N)) where {N<:Real}

Convert an empty set to a sequence of points for plotting. In the special case of an empty set, we define the sequence as nothing.

Input

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

Output

nothing.

plot_emptyset(∅::EmptySet, [ε]::N=zero(N); ...)

Plot an empty set.

Input

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