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Centrally symmetric polytopes (AbstractCentrallySymmetricPolytope)

A centrally symmetric polytope is a combination of two other interfaces: Centrally symmetric sets and Polytope.

LazySets.AbstractCentrallySymmetricPolytope — Type
AbstractCentrallySymmetricPolytope{N} <: AbstractPolytope{N}

Abstract type for centrally symmetric, polytopic sets. It combines the AbstractCentrallySymmetric and AbstractPolytope interfaces. Such a type combination is necessary as long as Julia does not support multiple inheritance.

Notes

Every concrete AbstractCentrallySymmetricPolytope must define the following methods:

  • from AbstractCentrallySymmetric:
    • center(::AbstractCentrallySymmetricPolytope) – return the center point
    • center(::AbstractCentrallySymmetricPolytope, i::Int) – return the center point at index i
  • from AbstractPolytope:
    • vertices_list(::AbstractCentrallySymmetricPolytope) – return a list of all vertices

The subtypes of AbstractCentrallySymmetricPolytope (including abstract interfaces):

julia> subtypes(AbstractCentrallySymmetricPolytope)
2-element Vector{Any}:
 AbstractZonotope
 Ball1
source

This interface defines the following functions:

LazySets.API.an_element — Method
an_element(P::AbstractCentrallySymmetricPolytope)

Return some element of a centrally symmetric, polytopic set.

Input

  • P – centrally symmetric, polytopic set

Output

The center of the centrally symmetric, polytopic set.

source
LazySets.API.dim — Method
dim(P::AbstractCentrallySymmetricPolytope)

Return the ambient dimension of a centrally symmetric, polytopic set.

Input

  • P – centrally symmetric, polytopic set

Output

The ambient dimension of the polytopic set.

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LazySets.API.center — Method
center(S::AbstractCentrallySymmetricPolytope, i::Int)

Return the center of a centrally symmetric, polytopic set along a given dimension.

Input

  • S – centrally symmetric, polytopic set
  • i – dimension of interest

Output

The center along the given dimension.

source
Base.extrema — Method
extrema(S::AbstractCentrallySymmetricPolytope)

Return two vectors with the lowest and highest coordinate of a centrally symmetric, polytopic set.

Input

  • S – centrally symmetric, polytopic set

Output

Two vectors with the lowest and highest coordinates of S.

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(S::AbstractCentrallySymmetricPolytope, i::Int)

Return the lower and higher coordinate of a centrally symmetric, polytopic set in a given dimension.

Input

  • S – centrally symmetric, polytopic set
  • i – dimension of interest

Output

The lower and higher coordinate of the centrally symmetric, polytopic set in the given dimension.

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
Base.isempty — Method
isempty(P::AbstractCentrallySymmetricPolytope)

Check whether a centrally symmetric, polytopic set is empty.

Input

  • P – centrally symmetric, polytopic set

Output

false.

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LazySets.API.isuniversal — Function
isuniversal(S::AbstractCentrallySymmetricPolytope, [witness]::Bool=false)

Check whether a centrally symmetric, polytopic set is universal.

Input

  • S – centrally symmetric, polytopic 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 $v ∉ S$

Algorithm

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

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Implementations