diff --git a/Convex_hull_3/include/CGAL/convexity_check_3.h b/Convex_hull_3/include/CGAL/convexity_check_3.h
index 929001806cc..cd18bf1d394 100644
--- a/Convex_hull_3/include/CGAL/convexity_check_3.h
+++ b/Convex_hull_3/include/CGAL/convexity_check_3.h
@@ -187,15 +187,15 @@ bool is_strongly_convex_3(const Polyhedron& P, const Traits& traits)
 }
 
 template<class Polyhedron, class R>
-bool CGAL_is_strongly_convex_3(Polyhedron& P, Point_3<R>*)
+bool CGAL_is_strongly_convex_3(const Polyhedron& P, Point_3<R>*)
 {
   return is_strongly_convex_3(P, R());
 }
 
 template<class Polyhedron>
-bool is_strongly_convex_3(Polyhedron& P)
+bool is_strongly_convex_3(const Polyhedron& P)
 {
-  typedef typename boost::property_map<Polyhedron, vertex_point_t>::type Ppmap;
+  typedef typename boost::property_map<Polyhedron, vertex_point_t>::const_type Ppmap;
   typedef typename boost::property_traits<Ppmap>::value_type Point_3;
 
   return CGAL_is_strongly_convex_3(P, reinterpret_cast<Point_3*>(0));
@@ -204,7 +204,7 @@ bool is_strongly_convex_3(Polyhedron& P)
 template <class ForwardIterator, class Polyhedron, class Traits>
 bool all_points_inside( ForwardIterator first,
                         ForwardIterator last,
-                        Polyhedron& P,
+                        const Polyhedron& P,
                         const Traits&  traits)
 {
   typedef typename Traits::Plane_3                           Plane_3;