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Centrally symmetric sets (AbstractCentrallySymmetric)

Centrally symmetric convex sets such as balls of different norms are characterized by a center. Note that there is a special interface combination Centrally symmetric polytope.

LazySets.AbstractCentrallySymmetric — Type
AbstractCentrallySymmetric{N} <: ConvexSet{N}

Abstract type for centrally symmetric compact convex sets.

Notes

Every concrete AbstractCentrallySymmetric must define the following function:

  • center(::AbstractCentrallySymmetric) – return the center point

The subtypes of AbstractCentrallySymmetric:

julia> subtypes(AbstractCentrallySymmetric)
2-element Vector{Any}:
 AbstractBallp
 Ellipsoid
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This interface requires to implement the following function:

LazySets.API.center — Method
center(X::LazySet)

Compute the center of a centrally symmetric set.

Input

  • X – centrally symmetric set

Output

A vector with the center, or midpoint, of X.

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This interface defines the following functions:

LazySets.API.an_element — Method
an_element(X::LazySet)

Return some element of a nonempty set.

Input

  • X – set

Output

An element of X unless X is empty.

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Base.extrema — Method
extrema(X::LazySet)

Compute the lowest and highest coordinate of a set in each dimension.

Input

  • X – set

Output

Two vectors with the lowest and highest coordinates of X in each dimension.

Notes

See also extrema(X::LazySet, i::Int).

The result is equivalent to (low(X), high(X)), but sometimes it can be computed more efficiently.

The resulting points are the lowest and highest corners of the box approximation, so they are not necessarily contained in X.

Algorithm

The default implementation computes the extrema via low and high.

source
Base.extrema — Method

Extended help

extrema(S::AbstractCentrallySymmetric)

Notes

The result is equivalent to (low(S), high(S)).

Algorithm

We compute high(S) and then compute the lowest coordinates with the help of center(S) (which is assumed to be cheaper to obtain).

source
Base.extrema — Method
extrema(X::LazySet, i::Int)

Compute the lowest and highest coordinate of a set in a given dimension.

Input

  • X – set
  • i – dimension

Output

Two real numbers representing the lowest and highest coordinate of the set in the given dimension.

Notes

The result is equivalent to (low(X, i), high(X, i)), but sometimes it can be computed more efficiently.

The resulting values are the lower and upper ends of the projection.

source
Base.extrema — Method

Extended help

extrema(S::AbstractCentrallySymmetric, i::Int)

Notes

The result is equivalent to (low(S, i), high(S, i)).

Algorithm

We compute high(S, i) and then compute the lowest coordinates with the help of center(S, i) (which is assumed to be cheaper to obtain).

source
LazySets.API.isuniversal — Method
isuniversal(X::LazySet, witness::Bool=false)

Check whether a set is universal.

Input

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

Output

  • If the witness option is deactivated: true iff $X = ℝ^n$
  • If the witness option is activated:
    • (true, []) iff $X = ℝ^n$
    • (false, v) iff $X ≠ ℝ^n$ for some $v ∉ X$
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LazySets.API.isuniversal — Function

Extended help

isuniversal(S::AbstractCentrallySymmetric, [witness]::Bool=false)

Algorithm

A witness is obtained by computing the support vector in direction d = [1, 0, …, 0] and adding d on top.

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Undocumented implementations:

Inherited from LazySet:

Inherited from ConvexSet:

Convenience functions

LazySets.○ — Method
○(c, a)

Convenience constructor of Ellipsoids or Ball2s depending on the type of a.

Input

  • c – center
  • a – additional parameter (either a shape matrix for Ellipsoid or a radius for Ball2)

Output

A Ellipsoids or Ball2s depending on the type of a.

Notes

"○" can be typed by \bigcirc<tab>.

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Implementations