#1176 - Box approximation for Rectification

src/Approximations/box_approximations.jl

@@ -41,6 +41,36 @@ box_approximation(S::AbstractHyperrectangle) =
 # special case: empty set
 box_approximation(∅::EmptySet) = ∅
 
+"""
+    box_approximation(r::Rectification{N}
+                     )::Union{Hyperrectangle{N}, EmptySet{N}} where {N<:Real}
+
+Overapproximate the rectification of a convex set by a tight hyperrectangle.
+
+### Input
+
+- `S` -- rectification of a convex set
+
+### Output
+
+A hyperrectangle.
+
+### Algorithm
+
+Box approximation and rectification distribute.
+Hence we first check whether the wrapped set is empty.
+If so, we return the empty set.
+Otherwise, we compute the box approximation of the wrapped set, rectify the
+resulting box (which is simple), and finally convert the resulting set to a box.
+"""
+function box_approximation(r::Rectification{N}
+                          )::Union{Hyperrectangle{N}, EmptySet{N}} where {N<:Real}
+    if isempty(r.X)
+        return EmptySet{N}()
+    end
+    return convert(Hyperrectangle, Rectification(box_approximation(r.X)))
+end
+
 """
     interval_hull
 

test/unit_box_approximation.jl

@@ -33,6 +33,11 @@ for N in [Float64, Rational{Int}, Float32]
     E = EmptySet{N}()
     @test box_approximation(E) == E
 
+    # rectification
+    @test box_approximation(Rectification(EmptySet{N}())) isa EmptySet{N}
+    r = Rectification(Ball1(N[0, 0], N(1)))
+    @test box_approximation(r) == Hyperrectangle(low=N[0, 0], high=N[1, 1])
+
     # ===================================================================
     # Testing box_approximation_symmetric (= symmetric interval hull)
     # ===================================================================