Polytopes: intersect, dual_cone, polarize

docs/src/PolyhedralGeometry/Polyhedra/auxiliary.md

@@ -31,6 +31,7 @@ lineality_space(P::Polyhedron)
 nfacets(P::Polyhedron)
 normalized_volume(P::Polyhedron)
 nvertices(P::Polyhedron)
+polarize(P::Polyhedron)
 print_constraints(A::AnyVecOrMat, b::AbstractVector; trivial::Bool = false)
 print_constraints(P::Polyhedron; trivial::Bool = false)
 rays(P::Polyhedron)

docs/src/PolyhedralGeometry/cones.md

@@ -63,6 +63,8 @@ f_vector(C::Cone)
 hilbert_basis(C::Cone)
 codim(C::Cone)
 dim(C::Cone)
+dual_cone(C::Cone)
+intersect(C0::Cone, C1::Cone)
 ispointed(C::Cone)
 isfulldimensional(C::Cone)
 lineality_dim(C::Cone)

src/Polytopes/Cone/standard_constructions.jl

@@ -0,0 +1,56 @@
+###############################################################################
+###############################################################################
+### Standard constructions
+###############################################################################
+###############################################################################
+@doc Markdown.doc"""
+    intersect(C0::Cone, C1::Cone)
+
+Return the intersection $C0 \cap C1$ of `C0` and `C1`.
+
+# Examples
+```jldoctest
+julia> C0 = positive_hull([1 0])
+A polyhedral cone in ambient dimension 2
+
+julia> C1 = positive_hull([0 1])
+A polyhedral cone in ambient dimension 2
+
+julia> C01 = intersect(C0, C1)
+A polyhedral cone in ambient dimension 2
+
+julia> rays(C01)
+0-element VectorIterator{RayVector{Polymake.Rational}}
+
+julia> dim(C01)
+0
+```
+"""
+function intersect(C0::Cone, C1::Cone)
+   return Cone(Polymake.polytope.intersection(pm_cone(C0), pm_cone(C1)))
+end
+
+
+@doc Markdown.doc"""
+    dual_cone(C::Cone)
+
+Return the dual cone of `C` consisting of all those linear functions that
+evaluate positively on all o `C`.
+
+# Examples
+```jldoctest
+julia> C = positive_hull([1 0; -1 2])
+A polyhedral cone in ambient dimension 2
+
+julia> Cv = dual_cone(C)
+A polyhedral cone in ambient dimension 2
+
+julia> rays(Cv)
+2-element VectorIterator{RayVector{Polymake.Rational}}:
+ [1, 1/2]
+ [0, 1]
+```
+"""
+function dual_cone(C::Cone)
+    return Cone(Polymake.polytope.polarize(pm_cone(C)))
+end

src/Polytopes/Polyhedron/standard_constructions.jl

@@ -681,3 +681,30 @@ Essentially, one can read the
 upperbounds.
 """
 upper_bound_theorem(d::Int,n::Int) = Polyhedron(Polymake.polytope.upper_bound_theorem(d,n))
+
+
+@doc Markdown.doc"""
+    polarize(P::Polyhedron)
+
+Return the polar dual of the polyhedron `P`, consisting of all linear functions
+whose absolute value on `P` does not exceed 1.
+
+# Examples
+```jldoctest
+julia> square = cube(2)
+A polyhedron in ambient dimension 2
+
+julia> P = polarize(square)
+A polyhedron in ambient dimension 2
+
+julia> vertices(P)
+4-element VectorIterator{PointVector{Polymake.Rational}}:
+ [1, 0]
+ [-1, 0]
+ [0, 1]
+ [0, -1]
+```
+"""
+function polarize(P::Polyhedron)
+    return Polyhedron(Polymake.polytope.polarize(pm_polytope(P)))
+end

src/Polytopes/Polytopes.jl

@@ -30,6 +30,7 @@ export Cone,
     cross,
     cube,
     dim,
+    dual_cone,
     faces,
     facets,
     facets_as_halfspace_matrix_pair,
@@ -82,6 +83,7 @@ export Cone,
     objective_function,
     orbit_polytope,
     point_matrix,
+    polarize,
     print_constraints,
     product,
     recession_cone,
@@ -109,6 +111,7 @@ include("iterators.jl")
 include("Polyhedron/constructors.jl")
 include("Cone/constructors.jl")
 include("Cone/properties.jl")
+include("Cone/standard_constructions.jl")
 include("Polyhedron/properties.jl")
 include("Polyhedron/standard_constructions.jl")
 include("PolyhedralFan/constructors.jl")