Extended functionality for existing set types

[schillic]

This issue collects open tasks for set types that occurred while revisiting them in the course of #3047. This includes:

• implementing a more efficient method to improve a less efficient fallback method
• implementing a new method that is currently missing
• improving an existing implementation

Ball1

• constructor with default center = origin (as for Ellipsoid)
• area
• volume
• radius(B, Inf) = B.radius (for other p there should be a default method to call)
• chebyshev_center_radius
• constraints_list: fix case for radius 0 and avoid allocations with Iterators.product

Ball2

• constructor with default center = origin (as for Ellipsoid)
• radius(B, Inf) = B.radius (for other p there should be a default method to call)
• distance

BallInf

• constructor with default center = origin (as for Ellipsoid)
• chebyshev_center_radius

Ballp

• constructor with default center = origin (as for Ellipsoid)
• area
• volume formula
• chebyshev_center_radius (for p > 2 it is just the same radius; for p < 2 there is a calculation with the norm of [1, …, 1])
• radius(B, Inf) = B.radius (for other p there should be a default method to call)

DensePolynomialZonotope

• minkowski_sum (Eq. (14) in [1])
• quadratic_map of a Zonotope (Theorem 1 in [1])
• overapproximate with a QuadraticMap (Corollary 1 in [1])
• reduce_order (Proposition 2 in [1])

[1] Reachability analysis of nonlinear systems using conservative polynomialization and non-convex sets

Ellipsoid

• area
• volume
• chebyshev_center_radius
• low/high
• distance

HParallelotope

• order (= return 1//1)
• generators
• σ
• ρ
• low/high
• center (could be more efficient to take the midpoint of low and high)
• ∈ (checking each constraint directly should be more efficient)
• vertices_list
• translate

Hyperplane

• _linear_map_hrep_helper
• _σ_hyperplane_halfspace with Val arguments

Hyperrectangle

• translate!

Line

• isuniversal (witness case)

Line2D

• ρ (reuse from Hyperplane, which gives Inf results instead of errors, also for other default implementations like low/high)
• normalize
• distance
• reflect(x, L) = _reflect_point_hyperplane(x, L.a, L.b)

LineSegment

• high
• low
• norm
• radius
• diameter
• chebyshev_center_radius
• rectify

RotatedHyperrectangle

• generators
• rand
• ∈ (same as LinearMap, share the method)
• low/high

VPolygon

• low/high/extrema with dimension argument

VPolytope

• low/high/extrema with dimension argument
• an_element (see VPolygon)

ZeroSet

• low/high/extrema with dimension argument

Zonotope

• split! (instead of passing two similar matrices and using copyto!, create a copy directly in split!; but this is a public function)
• generalized 2D vertices list (see FIXME)

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Bloating

• vertices_list (see constraints_list)
• ∈ (see constraints_list or use distance)
• isuniversal
• translate

Intersection

• ρ_helper: revisit whether isempty is reasonable (because it is expensive the first time)

all lazy operations

• is_polyhedral (see also #3158)

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AbstractSingleton

• extrema (2x)
• order
• radius
• norm
• ρ(d::SingleEntryVector, H) (generalize existing methods for H::Hyperrectangle and H::BallInf)

AbstractHyperrectangle

• vertices

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box_approximation

• box_approximation(::MinkowskiSum): use extrema and add the centers and radii
• turn specific methods into extrema methods for the set types instead
  • afterward let box_approximation(::ConvexHull) only use the "extrema" algorithm (and remove the algorithm argument)

isdisjoint

• Hyperplane/Hyperplane method: faster algorithm:
  • compare normal vectors (parallel or not); if equivalent, compare the offsets; otherwise they intersect