Approximation of Rectification of polytope

[schillic]

The claim below is invalid as shown by the counterexample Ball1{Float64,Array{Float64,1}}([0.546237073661734, 0.11805252160250493], 0.1432864241448218) (in blue) and the convex hull of the rectified vertices (dark).

The error is that intersections of facets with the "dimension hyperplanes" (the axes in 2D) become additional vertices.

---

Let P = CHull({x ∈ vertices(P)}) be a polytope. Then CHull(Rectification(P)) = CHull({rectify(x) | x ∈ vertices(P)}). In particular, vertices(CHull(Rectification(P))) ⊆ {rectify(x) | x ∈ vertices(P)} ⊆ CHull(Rectification(P)).

Here is an (invalid) implementation.

function vertices_list(R::Rectification; prune::Bool=true)
    vlist = rectify.(vertices_list(R.X))
    if prune
        convex_hull!(vlist)
    end
    return vlist
end

This can also be used for overapproximate(R::Rectification, VPolytope) = VPolytope(vertices_list(R)).

---

For easier development, the set can be plotted like this:

function concretize(R::Rectification)
    return LazySets.to_union_of_projections(R, true)
end

plot([X for X in array(concretize(R))])