diff --git a/src/sage/geometry/cone.py b/src/sage/geometry/cone.py
index fc0b3731945..6dabca389f9 100644
--- a/src/sage/geometry/cone.py
+++ b/src/sage/geometry/cone.py
@@ -211,7 +211,6 @@
 from sage.combinat.posets.posets import FinitePoset
 from sage.geometry.point_collection import PointCollection
 from sage.geometry.polyhedron.constructor import Polyhedron
-from sage.geometry.polyhedron.base import is_Polyhedron
 from sage.geometry.hasse_diagram import lattice_from_incidences
 from sage.geometry.toric_lattice import (ToricLattice, is_ToricLattice,
                                          is_ToricLatticeQuotient)
@@ -259,6 +258,9 @@ def is_Cone(x):
 
         sage: from sage.geometry.cone import is_Cone
         sage: is_Cone(1)
+        doctest:warning...
+        DeprecationWarning: is_Cone is deprecated, use isinstance instead
+        See https://github.com/sagemath/sage/issues/34307 for details.
         False
         sage: quadrant = Cone([(1,0), (0,1)])
         sage: quadrant
@@ -266,6 +268,8 @@ def is_Cone(x):
         sage: is_Cone(quadrant)
         True
     """
+    from sage.misc.superseded import deprecation
+    deprecation(34307, "is_Cone is deprecated, use isinstance instead")
     return isinstance(x, ConvexRationalPolyhedralCone)
 
 
@@ -439,7 +443,7 @@ def Cone(rays, lattice=None, check=True, normalize=True):
         0-d cone in 2-d lattice N
     """
     # Cone from Polyhedron
-    if is_Polyhedron(rays):
+    if isinstance(rays, sage.geometry.abc.Polyhedron):
         polyhedron = rays
         if lattice is None:
             lattice = ToricLattice(polyhedron.ambient_dim())
@@ -1894,7 +1898,7 @@ def cartesian_product(self, other, lattice=None):
             N+N(0, 1)
             in 2-d lattice N+N
         """
-        assert is_Cone(other)
+        assert isinstance(other, sage.geometry.abc.ConvexRationalPolyhedralCone)
         rc = super().cartesian_product(other, lattice)
         return ConvexRationalPolyhedralCone(rc.rays(), rc.lattice())
 
@@ -1952,7 +1956,7 @@ def __richcmp__(self, right, op):
             sage: c2 is c3
             False
         """
-        if is_Cone(right):
+        if isinstance(right, sage.geometry.abc.ConvexRationalPolyhedralCone):
             # We don't care about particular type of right in this case
             return richcmp((self.lattice(), self.rays()),
                            (right.lattice(), right.rays()), op)
@@ -2413,7 +2417,7 @@ def embed(self, cone):
             ValueError: 2-d cone in 3-d lattice N is not a face
             of 3-d cone in 3-d lattice N!
         """
-        assert is_Cone(cone)
+        assert isinstance(cone, sage.geometry.abc.ConvexRationalPolyhedralCone)
         if cone.ambient() is self:
             return cone
         if self.is_strictly_convex():
@@ -2935,7 +2939,7 @@ def facet_of(self):
         L = self._ambient._face_lattice_function()
         H = L.hasse_diagram()
         return self._sort_faces(
-            f for f in H.neighbors_out(L(self)) if is_Cone(f))
+            f for f in H.neighbors_out(L(self)) if isinstance(f, sage.geometry.abc.ConvexRationalPolyhedralCone))
 
     def facets(self):
         r"""
@@ -6387,9 +6391,8 @@ def random_cone(lattice=None, min_ambient_dim=0, max_ambient_dim=None,
     It's hard to test the output of a random process, but we can at
     least make sure that we get a cone back::
 
-        sage: from sage.geometry.cone import is_Cone
         sage: K = random_cone(max_ambient_dim=6, max_rays=10)
-        sage: is_Cone(K)
+        sage: isinstance(K, sage.geometry.abc.ConvexRationalPolyhedralCone)
         True
 
     The upper/lower bounds are respected::
diff --git a/src/sage/geometry/fan.py b/src/sage/geometry/fan.py
index 832a76bb65d..0fffcdeb58c 100644
--- a/src/sage/geometry/fan.py
+++ b/src/sage/geometry/fan.py
@@ -238,6 +238,8 @@
 from copy import copy
 from warnings import warn
 
+import sage.geometry.abc
+
 from sage.structure.richcmp import richcmp_method, richcmp
 from sage.combinat.combination import Combinations
 from sage.combinat.posets.posets import FinitePoset
@@ -245,7 +247,6 @@
                                 Cone,
                                 ConvexRationalPolyhedralCone,
                                 IntegralRayCollection,
-                                is_Cone,
                                 normalize_rays)
 from sage.geometry.hasse_diagram import lattice_from_incidences
 from sage.geometry.point_collection import PointCollection
@@ -571,7 +572,7 @@ def result():
         cones = ((), )
         rays = ()
         return result()
-    if is_Cone(cones[0]):
+    if isinstance(cones[0], sage.geometry.abc.ConvexRationalPolyhedralCone):
         # Construct the fan from Cone objects
         if lattice is None:
             lattice = cones[0].lattice()
@@ -749,11 +750,10 @@ def FaceFan(polytope, lattice=None):
         ValueError: face fans are defined only for
         polytopes containing the origin as an interior point!
     """
-    from sage.geometry.lattice_polytope import is_LatticePolytope
     interior_point_error = ValueError(
         "face fans are defined only for polytopes containing "
         "the origin as an interior point!")
-    if is_LatticePolytope(polytope):
+    if isinstance(polytope, sage.geometry.abc.LatticePolytope):
         if any(d <= 0 for d in polytope.distances([0] * polytope.dim())):
             raise interior_point_error
         cones = (f.ambient_vertex_indices() for f in polytope.facets())
@@ -843,8 +843,7 @@ def NormalFan(polytope, lattice=None):
     """
     dimension_error = ValueError(
         'the normal fan is only defined for full-dimensional polytopes')
-    from sage.geometry.lattice_polytope import is_LatticePolytope
-    if is_LatticePolytope(polytope):
+    if isinstance(polytope, sage.geometry.abc.LatticePolytope):
         if polytope.dim() != polytope.lattice_dim():
             raise dimension_error
         rays = polytope.facet_normals()
@@ -2425,7 +2424,7 @@ def embed(self, cone):
             ValueError: 2-d cone in 3-d lattice N does not belong
             to Rational polyhedral fan in 3-d lattice N!
         """
-        if not is_Cone(cone):
+        if not isinstance(cone, sage.geometry.abc.ConvexRationalPolyhedralCone):
             raise TypeError("%s is not a cone!" % cone)
         if cone.ambient() is self:
             return cone
diff --git a/src/sage/geometry/hyperplane_arrangement/hyperplane.py b/src/sage/geometry/hyperplane_arrangement/hyperplane.py
index 9ca1a7c42ac..dfd1e4c6169 100644
--- a/src/sage/geometry/hyperplane_arrangement/hyperplane.py
+++ b/src/sage/geometry/hyperplane_arrangement/hyperplane.py
@@ -108,9 +108,10 @@
 #                  http://www.gnu.org/licenses/
 # *****************************************************************************
 
+import sage.geometry.abc
 
-from sage.misc.cachefunc import cached_method
 from sage.geometry.linear_expression import LinearExpression, LinearExpressionModule
+from sage.misc.cachefunc import cached_method
 
 
 class Hyperplane(LinearExpression):
@@ -460,9 +461,8 @@ def intersection(self, other):
             sage: h.intersection(polytopes.cube())
             A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 3 vertices
         """
-        from sage.geometry.polyhedron.base import is_Polyhedron
         from sage.geometry.polyhedron.constructor import Polyhedron
-        if not is_Polyhedron(other):
+        if not isinstance(other, sage.geometry.abc.Polyhedron):
             try:
                 other = other.polyhedron()
             except AttributeError:
diff --git a/src/sage/geometry/lattice_polytope.py b/src/sage/geometry/lattice_polytope.py
index b652e68c946..666548b9f39 100644
--- a/src/sage/geometry/lattice_polytope.py
+++ b/src/sage/geometry/lattice_polytope.py
@@ -482,6 +482,9 @@ def is_LatticePolytope(x):
 
         sage: from sage.geometry.lattice_polytope import is_LatticePolytope
         sage: is_LatticePolytope(1)
+        doctest:warning...
+        DeprecationWarning: is_LatticePolytope is deprecated, use isinstance instead
+        See https://github.com/sagemath/sage/issues/34307 for details.
         False
         sage: p = LatticePolytope([(1,0), (0,1), (-1,-1)])
         sage: p                                                                         # needs palp
@@ -489,6 +492,8 @@ def is_LatticePolytope(x):
         sage: is_LatticePolytope(p)
         True
     """
+    from sage.misc.superseded import deprecation
+    deprecation(34307, "is_LatticePolytope is deprecated, use isinstance instead")
     return isinstance(x, LatticePolytopeClass)
 
 @richcmp_method
@@ -989,7 +994,7 @@ def _palp(self, command, reduce_dimension=False):
             sage: o = lattice_polytope.cross_polytope(3)
             sage: o._palp("poly.x -f")                                                  # needs palp
             'M:7 6 N:27 8 Pic:17 Cor:0\n'
-            sage: print(o._palp("nef.x -f -N -p")) # random time information            # needs palp
+            sage: print(o._palp("nef.x -f -N -p"))  # random time information           # needs palp
             M:27 8 N:7 6  codim=2 #part=5
             H:[0] P:0 V:2 4 5       0sec  0cpu
             H:[0] P:2 V:3 4 5       0sec  0cpu
@@ -1234,7 +1239,7 @@ def _read_nef_partitions(self, data):
 
             sage: o = lattice_polytope.cross_polytope(3)
             sage: s = o.nef_x("-p -N -Lv")                                              # needs palp
-            sage: print(s) # random time values                                         # needs palp
+            sage: print(s)  # random time values                                        # needs palp
             M:27 8 N:7 6  codim=2 #part=5
             3 6 Vertices in N-lattice:
                 1    0    0   -1    0    0
@@ -3139,7 +3144,7 @@ def normal_form(self, algorithm="palp_native", permutation=False):
             M( 12, -1, -9, -6,  6),
             M( 12, -1, -6, -3,  3)
             in 5-d lattice M
-            sage: P.normal_form(algorithm="palp_modified")  # not tested (22s; MemoryError on 32 bit), needs sage.groups
+            sage: P.normal_form(algorithm="palp_modified")      # not tested (22s; MemoryError on 32 bit), needs sage.groups
             M(  6,  0,  0,  0,  0),
             M( -6,  0,  0,  0,  0),
             M(  0,  1,  0,  0,  0),
@@ -5264,7 +5269,7 @@ def _read_nef_x_partitions(data):
 
         sage: o = lattice_polytope.cross_polytope(3)
         sage: s = o.nef_x("-N -p")                                                      # needs palp
-        sage: print(s) # random                                                         # needs palp
+        sage: print(s)  # random                                                        # needs palp
         M:27 8 N:7 6  codim=2 #part=5
          P:0 V:2 4 5       0sec  0cpu
          P:2 V:3 4 5       0sec  0cpu
diff --git a/src/sage/geometry/newton_polygon.py b/src/sage/geometry/newton_polygon.py
index 28101e70646..c8e185d01e6 100644
--- a/src/sage/geometry/newton_polygon.py
+++ b/src/sage/geometry/newton_polygon.py
@@ -14,6 +14,8 @@
 #                  https://www.gnu.org/licenses/
 #############################################################################
 
+import sage.geometry.abc
+
 from sage.structure.unique_representation import UniqueRepresentation
 from sage.structure.parent import Parent
 from sage.structure.element import Element
@@ -22,7 +24,6 @@
 
 from sage.rings.infinity import Infinity
 from sage.geometry.polyhedron.constructor import Polyhedron
-from sage.geometry.polyhedron.base import is_Polyhedron
 
 
 class NewtonPolygon_element(Element):
@@ -716,7 +717,7 @@ def _element_constructor_(self, arg, sort_slopes=True, last_slope=Infinity):
             sage: NewtonPolygon(1)
             Finite Newton polygon with 1 vertex: (0, 0)
         """
-        if is_Polyhedron(arg):
+        if isinstance(arg, sage.geometry.abc.Polyhedron):
             return self.element_class(arg, parent=self)
         if arg == 0:
             polyhedron = Polyhedron(base_ring=self.base_ring(), ambient_dim=2)
diff --git a/src/sage/geometry/polyhedral_complex.py b/src/sage/geometry/polyhedral_complex.py
index 09200a60be9..981404eb174 100644
--- a/src/sage/geometry/polyhedral_complex.py
+++ b/src/sage/geometry/polyhedral_complex.py
@@ -110,9 +110,11 @@
 # ****************************************************************************
 
 from copy import copy
+
+import sage.geometry.abc
+
 from sage.topology.cell_complex import GenericCellComplex
 from sage.geometry.polyhedron.constructor import Polyhedron
-from sage.geometry.polyhedron.base import is_Polyhedron
 from sage.modules.free_module_element import vector
 from sage.rings.integer_ring import ZZ
 from sage.graphs.graph import Graph
@@ -308,7 +310,7 @@ def __init__(self, maximal_cells=None, backend=None, maximality_check=True,
                 ambient_dim = next(iter(cells_dict[self._dim])).ambient_dim()
         self._ambient_dim = ambient_dim
         self._maximal_cells = cells_dict
-        if not all((is_Polyhedron(cell) and
+        if not all((isinstance(cell, sage.geometry.abc.Polyhedron) and
                    cell.ambient_dim() == self._ambient_dim)
                    for cell in self.maximal_cell_iterator()):
             raise ValueError("the given cells are not polyhedra " +
@@ -959,7 +961,7 @@ def __contains__(self, x):
             sage: (0, 0) in pc  # not a polyhedron
             False
         """
-        if not is_Polyhedron(x):
+        if not isinstance(x, sage.geometry.abc.Polyhedron):
             return False
         dim = x.dimension()
         return dim in self.cells() and x in self.cells()[dim]
@@ -2028,7 +2030,7 @@ def add_cell(self, cell):
         """
         if self._is_immutable:
             raise ValueError("this polyhedral complex is not mutable")
-        if not is_Polyhedron(cell) or cell.ambient_dim() != self._ambient_dim:
+        if not isinstance(cell, sage.geometry.abc.Polyhedron) or cell.ambient_dim() != self._ambient_dim:
             raise ValueError("the given cell is not a polyhedron " +
                              "in the same ambient space")
         # if cell is already in self, do nothing.
@@ -2191,7 +2193,7 @@ def remove_cell(self, cell, check=False):
         """
         if self._is_immutable:
             raise ValueError("this polyhedral complex is not mutable")
-        if not is_Polyhedron(cell) or cell.ambient_dim() != self._ambient_dim:
+        if not isinstance(cell, sage.geometry.abc.Polyhedron) or cell.ambient_dim() != self._ambient_dim:
             raise ValueError("the given cell is not a polyhedron " +
                              "in the same ambient space")
         # if cell is not in self, delete nothing.
diff --git a/src/sage/geometry/polyhedron/base.py b/src/sage/geometry/polyhedron/base.py
index f9ce84be09d..82e92b72bd3 100644
--- a/src/sage/geometry/polyhedron/base.py
+++ b/src/sage/geometry/polyhedron/base.py
@@ -74,10 +74,15 @@ def is_Polyhedron(X):
         sage: p = polytopes.hypercube(2)
         sage: from sage.geometry.polyhedron.base import is_Polyhedron
         sage: is_Polyhedron(p)
+        doctest:warning...
+        DeprecationWarning: is_Polyhedron is deprecated, use isinstance instead
+        See https://github.com/sagemath/sage/issues/34307 for details.
         True
         sage: is_Polyhedron(123456)
         False
     """
+    from sage.misc.superseded import deprecation
+    deprecation(34307, "is_Polyhedron is deprecated, use isinstance instead")
     return isinstance(X, Polyhedron_base)
 
 
diff --git a/src/sage/geometry/polyhedron/parent.py b/src/sage/geometry/polyhedron/parent.py
index 7dafad76437..5fad17dc55a 100644
--- a/src/sage/geometry/polyhedron/parent.py
+++ b/src/sage/geometry/polyhedron/parent.py
@@ -9,6 +9,8 @@
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
+import sage.geometry.abc
+
 from sage.structure.parent import Parent
 from sage.structure.element import get_coercion_model
 from sage.structure.unique_representation import UniqueRepresentation
@@ -22,8 +24,6 @@
 from sage.categories.fields import Fields
 from sage.categories.rings import Rings
 from sage.categories.modules import Modules
-
-from sage.geometry.polyhedron.base import is_Polyhedron
 from .representation import Inequality, Equation, Vertex, Ray, Line
 
 
@@ -691,7 +691,7 @@ def convert_base_ring_Hrep(lstlst):
             if convert and Vrep:
                 Vrep = [convert_base_ring(_) for _ in Vrep]
             return self.element_class(self, Vrep, Hrep, **kwds)
-        if nargs == 1 and is_Polyhedron(args[0]):
+        if nargs == 1 and isinstance(args[0], sage.geometry.abc.Polyhedron):
             copy = kwds.pop('copy', args[0].parent() is not self)
             mutable = kwds.pop('mutable', False)
 
diff --git a/src/sage/schemes/toric/variety.py b/src/sage/schemes/toric/variety.py
index aee10319916..6a6a8d3252a 100644
--- a/src/sage/schemes/toric/variety.py
+++ b/src/sage/schemes/toric/variety.py
@@ -2660,7 +2660,7 @@ def _orbit_closure_projection(self, cone, x):
         # TODO: make the following work nicely.
         # if x in cone.lattice():
         # return quot(x)
-        # assert is_Cone(x)
+        # assert x is ConvexRationalPolyhedralCone object
         # return Cone(x.rays(), lattice=quot)
 
     def orbit_closure(self, cone):