diff --git a/src/sage/geometry/cone.py b/src/sage/geometry/cone.py
index e7813040521..a4e11c5a9d3 100644
--- a/src/sage/geometry/cone.py
+++ b/src/sage/geometry/cone.py
@@ -1712,6 +1712,7 @@ def interior_contains(self, *args):
            point = point[0]
         return self._contains(point, 'interior')
 
+    @cached_method
     def interior(self):
         r"""
         Return the interior of ``self``.
@@ -1761,11 +1762,12 @@ def relative_interior_contains(self, *args):
            point = point[0]
         return self._contains(point, 'relative interior')
 
+    @cached_method
     def relative_interior(self):
         r"""
         Return the relative interior of ``self``.
         """
-        if self.is_full_space():
+        if self.is_trivial() or self.is_full_space():
             return self
         return RelativeInterior(self)
 
diff --git a/src/sage/geometry/polyhedron/base.py b/src/sage/geometry/polyhedron/base.py
index 1175d21925f..985a9f5c6c5 100644
--- a/src/sage/geometry/polyhedron/base.py
+++ b/src/sage/geometry/polyhedron/base.py
@@ -8500,8 +8500,15 @@ def relative_interior(self):
             sage: (1, 0) in ri_P
             False
 
+            sage: P0 = Polyhedron(vertices=[[1, 2]])
+            sage: P0.relative_interior() is P0
+            True
+
+            sage: Empty = Polyhedron(ambient_dim=2)
+            sage: Empty.relative_interior() is Empty
+            True
         """
-        if self.is_empty() or self.is_universe():
+        if self.dim() <= 0 or self.is_universe():
             return self
         return RelativeInterior(self)