diff --git a/Spatial_searching/doc/Spatial_searching/CGAL/Fuzzy_sphere.h b/Spatial_searching/doc/Spatial_searching/CGAL/Fuzzy_sphere.h
index 181ea1d312e..db3aec07db3 100644
--- a/Spatial_searching/doc/Spatial_searching/CGAL/Fuzzy_sphere.h
+++ b/Spatial_searching/doc/Spatial_searching/CGAL/Fuzzy_sphere.h
@@ -10,12 +10,12 @@ as outer approximation a sphere with radius \f$ r+\epsilon\f$.
 
 \attention The fuzziness of a `Fuzzy_sphere` is specified by a parameter \f$ \epsilon\f$
 denoting a maximal allowed distance to the boundary of a sphere.
-If the distance to the boundary is at least \f$ \epsilon\f$, points inside the
+If the distance to the boundary is greater than \f$ \epsilon\f$, points inside the
 object are always reported and points outside the sphere are never reported.
-Points within distance \f$ \epsilon\f$ to the boundary may be or may be not reported.
-Subsequently, points on the inner and outer spheres may or may not be reported
-by a search query. Specifically when \f$ \epsilon = 0\f$, points on the sphere
-of radius \f$ r\f$ may or may not be reported.
+Points whose distance to the boundary is less than or equal to \f$ \epsilon\f$
+may or may not be reported. Subsequently, points on the inner and outer spheres
+may or may not be reported. Specifically when \f$ \epsilon = 0\f$, points
+on the sphere of radius \f$ r\f$ may or may not be reported.
 
 \tparam Traits must be a model of the concept
 `SearchTraits`, for example `CGAL::Cartesian_d<double>`.
@@ -90,7 +90,7 @@ bool inner_range_intersects(const Kd_tree_rectangle<FT,D>& rectangle) const;
 /*!
 Test whether the outer sphere encloses the rectangle associated with a node of a tree.
 
-That is, whether the minimal distance between the center of the fuzzy sphere and
+That is, whether the maximal distance between the center of the fuzzy sphere and
 `rectangle` is less than \f$ r+\epsilon\f$.
 */
 bool outer_range_contains(const Kd_tree_rectangle<FT,D>& rectangle) const;
diff --git a/Spatial_searching/include/CGAL/Fuzzy_sphere.h b/Spatial_searching/include/CGAL/Fuzzy_sphere.h
index 7d2c01badbf..32439e65321 100644
--- a/Spatial_searching/include/CGAL/Fuzzy_sphere.h
+++ b/Spatial_searching/include/CGAL/Fuzzy_sphere.h
@@ -59,9 +59,7 @@ namespace CGAL {
 	}
         	
         bool contains(const typename SearchTraits::Point_d& p) const {
-		// test whether the squared distance 
-		// between P and c 
-		// is at most the squared_radius
+		// test whether the distance between c and p is less than the radius
 		FT squared_radius = r*r;
 		FT distance=FT(0);
 		typename SearchTraits::Construct_cartesian_const_iterator_d construct_it=
@@ -70,7 +68,7 @@ namespace CGAL {
 		                                                  pit = construct_it(p),
                                                                   end = construct_it(c, 0);
 		for (; cit != end
-                       && (distance <= squared_radius); ++cit, ++pit) {
+                       && (distance < squared_radius); ++cit, ++pit) {
 		  distance += 
 			((*cit)-(*pit))*((*cit)-(*pit));
 		}
@@ -78,14 +76,13 @@ namespace CGAL {
 		return (distance < squared_radius); 
         }
 
-        
-	bool inner_range_intersects(const Kd_tree_rectangle<FT,Dimension>& rectangle) const {                          
-                // test whether the interior of a sphere
-		// with radius (r-eps) intersects r, i.e.
-                // if the minimal distance of r to c is less than r-eps
-		FT distance = FT(0);
-		FT squared_radius = (r-eps)*(r-eps);
-		typename SearchTraits::Construct_cartesian_const_iterator_d construct_it=
+    bool inner_range_intersects(const Kd_tree_rectangle<FT,Dimension>& rectangle) const {
+      // test whether the interior of a sphere
+      // with radius (r-eps) intersects 'rectangle', i.e.
+      // if the minimal distance of c to 'rectangle' is less than r-eps
+      FT distance = FT(0);
+      FT squared_radius = (r-eps)*(r-eps);
+      typename SearchTraits::Construct_cartesian_const_iterator_d construct_it=
                   traits.construct_cartesian_const_iterator_d_object();
                 typename SearchTraits::Cartesian_const_iterator_d cit = construct_it(c),
                                                                   end = construct_it(c, 0);
@@ -102,17 +99,17 @@ namespace CGAL {
 	}
 
 
-	bool outer_range_contains(const Kd_tree_rectangle<FT,Dimension>& rectangle) const { 
-        // test whether the interior of a sphere
-	// with radius (r+eps) is contained by r, i.e.
-        // if the minimal distance of the boundary of r 
-        // to c is less than r+eps                         
-	FT distance=FT(0);
-	FT squared_radius = (r+eps)*(r+eps);	
-	typename SearchTraits::Construct_cartesian_const_iterator_d construct_it=
+    bool outer_range_contains(const Kd_tree_rectangle<FT,Dimension>& rectangle) const { 
+      // test whether the interior of a sphere
+      // with radius (r+eps) is contained by 'rectangle', i.e.
+      // if the maximal distance of c to the boundary of 'rectangle'
+      // is less than r+eps
+      FT distance=FT(0);
+      FT squared_radius = (r+eps)*(r+eps);	
+      typename SearchTraits::Construct_cartesian_const_iterator_d construct_it=
           traits.construct_cartesian_const_iterator_d_object();
-	typename SearchTraits::Cartesian_const_iterator_d cit = construct_it(c),
-                                                          end = construct_it(c, 0);
+      typename SearchTraits::Cartesian_const_iterator_d cit = construct_it(c),
+                                                        end = construct_it(c, 0);
         for (int i = 0; cit != end && (distance < squared_radius) ; ++cit,++i) {
 		if ((*cit) <= (rectangle.min_coord(i)+rectangle.max_coord(i))/FT(2))
 			distance +=