diff --git a/Convex_hull_3/include/CGAL/Convex_hull_3/dual/halfspace_intersection_3.h b/Convex_hull_3/include/CGAL/Convex_hull_3/dual/halfspace_intersection_3.h
index 15cf91f451a..7ea9e73a8ad 100644
--- a/Convex_hull_3/include/CGAL/Convex_hull_3/dual/halfspace_intersection_3.h
+++ b/Convex_hull_3/include/CGAL/Convex_hull_3/dual/halfspace_intersection_3.h
@@ -243,7 +243,7 @@ namespace CGAL
     template <class PlaneIterator, class Polyhedron>
     void halfspace_intersection_3 (PlaneIterator begin, PlaneIterator end,
                                    Polyhedron &P,
-                                   boost::optional<typename Polyhedron::Vertex::Point_3> const& origin = boost::none) {
+                                   boost::optional<typename Polyhedron::Vertex::Point_3> origin = boost::none) {
         // Checks whether the intersection is a polyhedron
         CGAL_assertion_msg(Convex_hull_3::internal::is_intersection_dim_3(begin, end), "halfspace_intersection_3: intersection not a polyhedron");
 
@@ -252,20 +252,9 @@ namespace CGAL
         typedef Convex_hull_3::Convex_hull_traits_dual_3<K> Hull_traits_dual_3;
         typedef Polyhedron_3<Hull_traits_dual_3> Polyhedron_dual_3;
         typedef Convex_hull_3::internal::Build_primal_polyhedron<K, Polyhedron_dual_3, Polyhedron> Builder;
-        typedef typename Polyhedron::Vertex::Point_3 Point_3;
 
-        if (origin) {
-            Point_3 p_origin = boost::get(origin);
-            Hull_traits_dual_3 dual_traits(p_origin);
-
-            Polyhedron_dual_3 dual_convex_hull;
-            CGAL::convex_hull_3(begin, end, dual_convex_hull, dual_traits);
-            Builder build_primal(dual_convex_hull, p_origin);
-            P.delegate(build_primal);
-
-            // Posterior check if the origin is inside the computed polyhedron
-            CGAL_assertion_msg(Convex_hull_3::internal::point_inside_convex_polyhedron(P, p_origin), "halfspace_intersection_3: origin not in the polyhedron");
-        } else {
+        // if a point inside is not provided find one using linear programming
+        if (!origin) {
           // choose exact integral type
 #ifdef CGAL_USE_GMP
           typedef CGAL::Gmpq ET;
@@ -278,18 +267,18 @@ namespace CGAL
           CGAL_assertion_code(bool interior_point_found = )
           interior.find(begin, end);
           CGAL_assertion_msg(interior_point_found, "halfspace_intersection_3: problem when determing a point inside the intersection");
-          Point_3 origin = interior.inside_point();
-
-          Hull_traits_dual_3 dual_traits(origin);
-
-          Polyhedron_dual_3 dual_convex_hull;
-          CGAL::convex_hull_3(begin, end, dual_convex_hull, dual_traits);
-          Builder build_primal(dual_convex_hull, origin);
-          P.delegate(build_primal);
-
-          // Posterior check if the origin is inside the computed polyhedron
-          CGAL_assertion_msg(Convex_hull_3::internal::point_inside_convex_polyhedron(P, origin), "halfspace_intersection_3: origin not in the polyhedron");
+          origin = boost::make_optional(interior.inside_point());
         }
+
+        // compute the intersection of the half-space using the dual formulation
+        Hull_traits_dual_3 dual_traits(*origin);
+        Polyhedron_dual_3 dual_convex_hull;
+        CGAL::convex_hull_3(begin, end, dual_convex_hull, dual_traits);
+        Builder build_primal(dual_convex_hull, *origin);
+        P.delegate(build_primal);
+
+        // Posterior check if the origin is inside the computed polyhedron
+        CGAL_assertion_msg(Convex_hull_3::internal::point_inside_convex_polyhedron(P, *origin), "halfspace_intersection_3: origin not in the polyhedron");
     }
 
   #ifndef CGAL_NO_DEPRECATED_CODE