Operations on sets

In this section we show which typical set operations this library supports.

We use the following four sets for illustration.

using LazySets, LazySets.Approximations, Plots
B1 = Ball1(-ones(2), 1.)
B2 = Ball2(ones(2), 1.)
BI = BallInf(zeros(2), 1.)
H = Hyperrectangle(ones(2), ones(2))
sets = [B1, B2, BI, H]

function plot_sets(sets)
    for S in sets
        println(S)
        plot!(S, 1e-2, fillalpha=0.1)
    end
end

function plot_points(points, prefix)
    for i in eachindex(points)
        p = points[i]
        num_occur = length(find(x -> x == p, points[1:i]))
        x = p[1]
        y = p[2]
        if num_occur == 1
            x += 0.15
        elseif num_occur == 2
            y += 0.15
        elseif num_occur == 3
            x -= 0.15
        else
            y -= 0.15
        end
        plot!(Singleton(p))
        plot!(annotations=(x, y, text("$(prefix)$(i)")))
    end
end

plot1 = plot()
plot_sets(sets)
plot1

Unary operations

The following table lists all operations (apart from dim and σ, which must be supported by all types) that take one convex set as argument in the columns. In the rows we list all set types, both the interfaces (where we abbreviate the Abstract prefix), the basic set types, and the lazy set operations, each sorted alphabetically. The table entries have the following meaning.

An "x" indicates that the operation is implemented for the respective set.
"(⋅)" indicates that the operation is inherited from a supertype.

type ↓ \ operation →	radius	diameter	norm	an_element	∈
Interfaces
`LazySet`
`AHPolygon`				x	x
`AHyperrectangle`	x	x	x	(x)	x
`APointSymmetric`				x
`APointSymmetricPolytope`				x
`APolygon`
`APolytope`
`ASingleton`	(x)	(x)	(x)	x	x

Basic set types
`Ball1`				(x)	x
`Ball2`				(x)	x
`BallInf`	x	(x)	(x)	(x)	(x)
`Ballp`				(x)	x
`Ellipsoid`				(x)	x
`EmptySet`				x	x
`HPolygon`/`HPolygonOpt`				(x)	(x)
`HPolytope`					x
`Hyperrectangle`	(x)	(x)	(x)	(x)	(x)
`Singleton`	(x)	(x)	(x)	(x)	(x)
`VPolygon`				x	x
`ZeroSet`	(x)	(x)	(x)	(x)	x
`Zonotope`				(x)	x

Lazy set operation types
`CartesianProduct`					x
`CartesianProductArray`					x
`ConvexHull`
`ConvexHullArray`
`ExponentialMap`					x
`ExponentialProjectionMap`
`HalfSpace`				x	x
`Hyperplane`				x	x
`Intersection`					x
`LinearMap`					x
`MinkowskiSum`
`MinkowskiSumArray`
`SymmetricIntervalHull`	(x)	(x)	(x)	(x)	(x)

`radius`

This function returns the radius of a set. It is defined as the radius of the enclosing ball (of the given norm) of minimal volume with the same center.

radius(B1), radius(B2), radius(BI), radius(H)

(1.0, 1.0, 1.0, 1.0)

`diameter`

This function returns the diameter of a set. It is defined as the diameter of the enclosing ball (of the given norm) of minimal volume with the same center. The implementation is inherited for all set types if the norm is the infinity norm, in which case the result is defined as twice the radius.

diameter(B1), diameter(B2), diameter(BI), diameter(H)

(2.0, 2.0, 2.0, 2.0)

`norm`

This function returns the norm of a set. It is defined as the norm of the enclosing ball (of the given norm) of minimal volume centered in the origin.

# print 1-norm, 2-norm, and infinity norm (if available)
println(("-", "-", norm(B1, Inf)))
println(("-", "-", norm(B2, Inf)))
println((norm(BI, 1), norm(BI, 2), norm(BI, Inf)))
println((norm(H, 1), norm(H, 2), norm(H, Inf)))

("-", "-", 2.0)
("-", "-", 2.0)
(2.0, 1.4142135623730951, 1.0)
(4.0, 2.8284271247461903, 2.0)

`an_element`

This function returns some element in the set. Consecutive calls to this function typically return the same element.

an_element(B1), an_element(B2), an_element(BI), an_element(H)

([-1.0, -1.0], [1.0, 1.0], [0.0, 0.0], [1.0, 1.0])

`∈`

This function checks containment of a given vector in the set. The operator can be used in infix notation (v ∈ S) and in inverse operand order (S ∋ v). Alias: in

p1 = [1.5, 1.5]
p2 = [0.1, 0.1]
p3 = [-0.9, -0.8]
points = [p1, p2, p3]

for p in [p1, p2, p3]
    println("$p ∈ (B1, B2, BI, H)? ($(p ∈ B1), $(p ∈ B2), $(p ∈ BI), $(p ∈ H))")
end

[1.5, 1.5] ∈ (B1, B2, BI, H)? (false, true, false, true)
[0.1, 0.1] ∈ (B1, B2, BI, H)? (false, false, true, true)
[-0.9, -0.8] ∈ (B1, B2, BI, H)? (true, false, true, false)

plot1 = plot()
plot_sets(sets)
plot_points(points, "p")
plot1

Binary operations

The following table lists all operations that take two convex set as argument in the entries. In the rows we list all set types, both the interfaces (where we abbreviate the Abstract prefix), the basic set types, and the lazy set operations, each sorted alphabetically. In the columns we also list the operations, but abbreviated. The table entries consist of subsets of the following list of operations.

"⊆" stands for the subset check operation.
"∩=∅" stands for the intersection emptiness check operation.
"(⋅)" indicates that the operation is inherited from a supertype.
"[⋅]" indicates that a possible type ambiguity can occur but is resolved.
"{⋅}" indicates that a more efficient implementation is used instead of the inherited one.

type ↓ \ type →	LazyS	AHPon	AHrec	APSym	APSPol	APgon	APtop	ASingle	B1	B2	BInf	Bp	Ellip	EmpSt	HPgon	HPtop	Hrect	Single	VPgon	ZeroSet	Zonot	CP	CPA	CH	CHA	EMap	EPM	HalfS	HypPl	Inter	LMap	MSum	MSA	SIH
Interfaces
`LazySet`			⊆					(⊆),∩=∅			(⊆)						(⊆)	(⊆)		(⊆)														(⊆)
`AHPolygon`	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)
`AHyperrectangle`	(⊆)	(⊆)	{⊆},∩=∅	(⊆)	(⊆)	(⊆)	(⊆)	[⊆,∩=∅]	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)
`APointSymmetric`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`APointSymmetricPolytope`	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)
`APolygon`	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)
`APolytope`	⊆	(⊆)	[⊆]	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)
`ASingleton`	{⊆},∩=∅	(⊆,∩=∅)	[⊆,∩=∅]	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	[⊆,∩=∅]	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	((⊆,∩=∅))	(⊆,∩=∅)	((⊆,∩=∅))	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)

Basic set types
`Ball1`	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)
`Ball2`			(⊆)					{⊆},(∩=∅)		⊆,∩=∅								(⊆,∩=∅)		(⊆,∩=∅)
`BallInf`	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)
`Ballp`			(⊆)					{⊆},(∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`Ellipsoid`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`EmptySet`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`HPolygon`/`HPolygonOpt`	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)
`HPolytope`	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)
`Hyperrectangle`	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)
`Singleton`	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)
`VPolygon`	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)
`ZeroSet`	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)	(⊆,∩=∅)
`Zonotope`	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)

Lazy set operation types
`CartesianProduct`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`CartesianProductArray`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`ConvexHull`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`ConvexHullArray`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`ExponentialMap`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`ExponentialProjectionMap`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`HalfSpace`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`Hyperplane`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`Intersection`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`LinearMap`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`MinkowskiSum`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`MinkowskiSumArray`			(⊆)					(⊆,∩=∅)										(⊆,∩=∅)		(⊆,∩=∅)
`SymmetricIntervalHull`	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆,∩=∅)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)	(⊆)

`⊆`

This function checks whether a set is a subset of another set. It can optionally produce a witness if the subset relation does not hold. The operator can be used in infix notation (X ⊆ S). Alias: issubset

println(B1 ⊆ B2)
w1 = ⊆(B1, B2, true)[2]
println(B1 ⊆ BI)
w2 = ⊆(B1, BI, true)[2]
println(B1 ⊆ H)
w3 = ⊆(B1, H, true)[2]
# 'B2 ⊆ B1' is not supported yet
# w11 = ⊆(B2, B1, true)[2]
println(B2 ⊆ BI)
w4 = ⊆(B2, BI, true)[2]
println(B2 ⊆ H)
println(BI ⊆ B1)
w5 = ⊆(BI, B1, true)[2]
println(BI ⊆ B2)
w6 = ⊆(BI, B2, true)[2]
println(BI ⊆ H)
w7 = ⊆(BI, H, true)[2]
println(H ⊆ B1)
w8 = ⊆(H, B1, true)[2]
println(H ⊆ B2)
w9 = ⊆(H, B2, true)[2]
println(H ⊆ BI)
w10 = ⊆(H, BI, true)[2];

false
false
false
false
true
false
false
false
false
false
false

witnesses = [w1, w2, w3, w4, w5, w6, w7, w8, w9, w10]

plot1 = plot()
plot_sets(sets)
plot_points(witnesses, "w")
plot1

`is_intersection_empty`

This function checks whether the intersection of two sets is empty. It can optionally produce a witness if the intersection is nonempty.

println(is_intersection_empty(BI, H))
w1 = is_intersection_empty(BI, H, true)[2]
# none of the other combinations are supported yet
# is_intersection_empty(B1, B2)
# is_intersection_empty(B1, BI)
# is_intersection_empty(B1, H)
# w2 = is_intersection_empty(B1, H, true)[2]
# is_intersection_empty(B2, BI)
# is_intersection_empty(B2, H)

false

witnesses = [w1]

plot1 = plot()
plot_sets(sets)
plot_points(witnesses, "w")
plot1