diff --git a/src/sage/geometry/convex_set.py b/src/sage/geometry/convex_set.py
index c6dc7967e4d..ef9f58b2222 100644
--- a/src/sage/geometry/convex_set.py
+++ b/src/sage/geometry/convex_set.py
@@ -443,6 +443,45 @@ def affine_hull(self):
             TypeError: 'NotImplementedType' object is not callable
         """
 
+    def an_element(self):
+        r"""
+        Return a point of ``self``.
+
+        If ``self`` is empty, an :class:`EmptySetError` will be raised.
+
+        The default implementation delegates to :meth:`some_elements`.
+
+        EXAMPLES::
+
+            sage: from sage.geometry.convex_set import ConvexSet_compact
+            sage: class BlueBox(ConvexSet_compact):
+            ....:     def some_elements(self):
+            ....:         yield 'blue'
+            ....:         yield 'cyan'
+            sage: BlueBox().an_element()
+        """
+        try:
+            return next(iter(self.some_elements()))
+        except StopIteration:
+            raise EmptySetError
+
+    @abstract_method(optional=True)
+    def some_elements(self):
+        r"""
+        Generate some points of ``self``.
+
+        If ``self`` is empty, no points are generated; no exception will be raised.
+
+        TESTS::
+
+            sage: from sage.geometry.convex_set import ConvexSet_base
+            sage: C = ConvexSet_base()
+            sage: C.some_elements(C)
+            Traceback (most recent call last):
+            ...
+            TypeError: 'NotImplementedType' object is not callable
+        """
+
     @abstract_method(optional=True)
     def cartesian_product(self, other):
         """
diff --git a/src/sage/geometry/polyhedron/base.py b/src/sage/geometry/polyhedron/base.py
index 82af7e683fc..ac1a12fa2e1 100644
--- a/src/sage/geometry/polyhedron/base.py
+++ b/src/sage/geometry/polyhedron/base.py
@@ -3302,6 +3302,35 @@ def representative_point(self):
         accumulator.set_immutable()
         return accumulator
 
+    def some_elements(self):
+        r"""
+        Generate some points of ``self``.
+
+        If ``self`` is empty, no points are generated; no exception will be raised.
+
+        EXAMPLES::
+
+            sage: P = polytopes.simplex()
+            sage: list(P.some_elements())
+            [(1/4, 1/4, 1/4, 1/4),
+             (0, 0, 0, 1),
+             (0, 0, 1/2, 1/2),
+             (0, 1/2, 1/4, 1/4),
+             (1/2, 1/4, 1/8, 1/8)]
+        """
+        if self.is_empty():
+            return
+        yield self.representative_point()
+        vertex_iter = iter(self.vertex_generator())
+        try:
+            p = next(vertex_iter).vector()
+            yield vector(p, immutable=True)
+            for i in range(4):
+                p = (p + next(vertex_iter).vector()) / 2
+                yield vector(p, immutable=True)
+        except StopIteration:
+            pass
+
     def a_maximal_chain(self):
         r"""
         Return a maximal chain of the face lattice in increasing order.
diff --git a/src/sage/geometry/relative_interior.py b/src/sage/geometry/relative_interior.py
index c714faabcb9..b617ff39697 100644
--- a/src/sage/geometry/relative_interior.py
+++ b/src/sage/geometry/relative_interior.py
@@ -227,6 +227,22 @@ def is_closed(self):
         assert not self._polyhedron.is_relatively_open()
         return False
 
+    def some_elements(self):
+        r"""
+        Generate some points of ``self``.
+
+        If ``self`` is empty, no points are generated; no exception will be raised.
+
+        EXAMPLES::
+
+            sage: P = polytopes.simplex()
+            sage: list(P.relative_interior().some_elements())
+            [(1/4, 1/4, 1/4, 1/4), (1/2, 1/4, 1/8, 1/8)]
+        """
+        for p in self._polyhedron.some_elements():
+            if p in self:
+                yield p
+
     def _repr_(self):
         r"""
         Return a description of ``self``.