#619 Is intersection empty for polytopes

[mforets]

Closes #619.

[schillic]

I am not sure if this is the correct interpretation. To me, an HPolytope/HPolygon with no constraints is either not defined or universal (because each new constraint removes parts of the set).

Also, add a new line at the end of the file.

[mforets]

I agree that the HPolytope with no constraints is particular case that could be treated separately as being "universal", the whole R^n.

For VPolytope, i think it is reasonable to say that if it has no vertices then it only contains the empty set, V = convex_hull{{}}.

I was concerned with the case where the set is defined by a set of constraints with empty intersection. For instance consider the pair of constraints x <= -1 && x >= 1:

julia> P = HPolytope([LazySets.HalfSpace([1.0, 0.0], -1.0), LazySets.HalfSpace([-1.0, 0.0], -1.0)])
LazySets.HPolytope{Float64}(LazySets.HalfSpace{Float64}[LazySets.HalfSpace{Float64}([1.0, 0.0], -1.0), LazySets.HalfSpace{Float64}([-1.0, 0.0], -1.0)])

julia> isempty(P)
true

julia> vertices_list(P)
0-element Array{Array{Float64,1},1}

[schillic]

Rather than finding workarounds, we could accept that our algorithms are generally not correct for invalid polytope inputs, like unbounded or contradicting constraints 😉

For VPolytope, i think it is reasonable to say that if it has no vertices then it only contains the empty set, V = convex_hull{{}}.

Sure.

If HPolytope and HPolygon are problematic, should we replace AbstractPolytope by Union{VPolygon, VPolytope} then? Our other polytopic sets can never be empty by definition anyway.

[schillic]

"a pair of" → "two"

[schillic]

LazySets.binary_operations:Test Failed
  Expression: check_method_ambiguity_binary(is_intersection_empty)

[mforets]

i have no clue about the failing doctest in 0.7

[schillic]

new line 😃

[mforets]

agghhh there is some obscure setting in my editor (atom) which removes the empty lines

[schillic]

i have no clue about the failing doctest in 0.7

You have to add the type parameter in interfaces.md:
isempty(::AbstractPolytope{N}) where {N<:Real}

Funny that it worked in one of the builds.