diff --git a/docs/tutorials/notebooks/basic_ot_between_datasets.ipynb b/docs/tutorials/notebooks/basic_ot_between_datasets.ipynb
index 3ce57c822..fffa06fd2 100644
--- a/docs/tutorials/notebooks/basic_ot_between_datasets.ipynb
+++ b/docs/tutorials/notebooks/basic_ot_between_datasets.ipynb
@@ -9,7 +9,7 @@
     "\n",
     "This short tutorial covers a basic use case for {mod}`ott`:\n",
     "\n",
-    "- Compute a optimal transport distance between two point clouds using the {class}`~ott.geometry.point_cloud.PointCloud` geometry, solved using the {class}`~ott.solvers.linear.sinkhorn.Sinkhorn` algorithm. \n",
+    "- Compute a optimal transport distance between two point clouds using the {class}`~ott.geometry.pointcloud.PointCloud` geometry, solved using the {class}`~ott.solvers.linear.sinkhorn.Sinkhorn` algorithm. \n",
     "- Showcase the seamless integration with `JAX`, to differentiate through that cost and plot the gradient flow to morph the first point cloud into the second."
    ]
   },
@@ -367,7 +367,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.10.8"
+   "version": "3.10.9"
   },
   "vscode": {
    "interpreter": {
diff --git a/src/ott/geometry/costs.py b/src/ott/geometry/costs.py
index 5e874f93b..e0712379a 100644
--- a/src/ott/geometry/costs.py
+++ b/src/ott/geometry/costs.py
@@ -175,17 +175,16 @@ def pairwise(self, x: jnp.ndarray, y: jnp.ndarray) -> float:
 
 @jax.tree_util.register_pytree_node_class
 class SqPNorm(TICost):
-  """Squared p-norm of the difference of two vectors.
+  r"""Squared p-norm of the difference of two vectors.
 
   Args:
-    p: Power of the p-norm.
+    p: Power of the p-norm, :math:`\ge 1`.
   """
 
   def __init__(self, p: float):
     super().__init__()
-    assert p >= 1.0, "p parameter in sq. p-norm should be >= 1.0"
     self.p = p
-    self.q = 1. / (1. - 1. / self.p) if p > 1.0 else jnp.inf
+    self.q = 1.0 / (1.0 - (1.0 / p)) if p > 1.0 else jnp.inf
 
   def h(self, z: jnp.ndarray) -> float:  # noqa: D102
     return 0.5 * jnp.linalg.norm(z, self.p) ** 2
@@ -203,29 +202,28 @@ def tree_flatten(self):  # noqa: D102
   @classmethod
   def tree_unflatten(cls, aux_data, children):  # noqa: D102
     del children
-    return cls(aux_data[0])
+    return cls(*aux_data)
 
 
 @jax.tree_util.register_pytree_node_class
 class PNormP(TICost):
-  """p-norm to the power p (and divided by p) of the difference of two vectors.
+  r"""p-norm to the power p (and divided by p) of the difference of two vectors.
 
   Args:
-    p: Power of the p-norm, a finite float larger than 1.0.
+    p: Power of the p-norm in :math:`[1, +\infty)`.
+      Note that :func:`h_legendre` is not defined for ``p = 1``.
   """
 
   def __init__(self, p: float):
     super().__init__()
-    assert p >= 1.0, "p parameter in p-norm should be larger than 1.0"
-    assert p < jnp.inf, "p parameter in p-norm should be finite"
     self.p = p
-    self.q = 1. / (1. - 1. / self.p) if p > 1.0 else jnp.inf
+    self.q = 1.0 / (1.0 - (1.0 / p)) if p > 1.0 else jnp.inf
 
   def h(self, z: jnp.ndarray) -> float:  # noqa: D102
     return jnp.linalg.norm(z, self.p) ** self.p / self.p
 
   def h_legendre(self, z: jnp.ndarray) -> float:  # noqa: D102
-    assert self.q < jnp.inf, "Legendre transform not defined for `p=1.0`"
+    # not defined for `p=1`
     return jnp.linalg.norm(z, self.q) ** self.q / self.q
 
   def tree_flatten(self):  # noqa: D102
@@ -234,7 +232,7 @@ def tree_flatten(self):  # noqa: D102
   @classmethod
   def tree_unflatten(cls, aux_data, children):  # noqa: D102
     del children
-    return cls(aux_data[0])
+    return cls(*aux_data)
 
 
 @jax.tree_util.register_pytree_node_class
@@ -337,12 +335,11 @@ class ElasticL1(RegTICost):
     \frac{1}{2} \|\cdot\|_2^2 + \gamma \|\cdot\|_1
 
   Args:
-    gamma: Strength of the :math:`\|\cdot\|_1` regularization.
+    gamma: Strength of the :math:`\|\cdot\|_1` regularization, :math:`\ge 0`.
   """
 
   def __init__(self, gamma: float = 1.0):
     super().__init__()
-    assert gamma >= 0, "Gamma must be non-negative."
     self.gamma = gamma
 
   def reg(self, z: jnp.ndarray) -> float:  # noqa: D102
@@ -352,12 +349,12 @@ def prox_reg(self, z: jnp.ndarray) -> float:  # noqa: D102
     return jnp.sign(z) * jax.nn.relu(jnp.abs(z) - self.gamma)
 
   def tree_flatten(self):  # noqa: D102
-    return (), (self.gamma,)
+    return (self.gamma,), None
 
   @classmethod
   def tree_unflatten(cls, aux_data, children):  # noqa: D102
-    del children
-    return cls(*aux_data)
+    del aux_data
+    return cls(*children)
 
 
 @jax.tree_util.register_pytree_node_class
@@ -373,12 +370,11 @@ class ElasticSTVS(RegTICost):
   where :math:`\sigma(\cdot) := \text{asinh}\left(\frac{\cdot}{2\gamma}\right)`
 
   Args:
-    gamma: Strength of the STVS regularization.
+    gamma: Strength of the STVS regularization, :math:`> 0`.
   """  # noqa
 
   def __init__(self, gamma: float = 1.0):
     super().__init__()
-    assert gamma > 0, "Gamma must be positive."
     self.gamma = gamma
 
   def reg(self, z: jnp.ndarray) -> float:  # noqa: D102
@@ -390,12 +386,12 @@ def prox_reg(self, z: jnp.ndarray) -> float:  # noqa: D102
     return jax.nn.relu(1 - (self.gamma / (jnp.abs(z) + 1e-12)) ** 2) * z
 
   def tree_flatten(self):  # noqa: D102
-    return (), (self.gamma,)
+    return (self.gamma,), None
 
   @classmethod
   def tree_unflatten(cls, aux_data, children):  # noqa: D102
-    del children
-    return cls(*aux_data)
+    del aux_data
+    return cls(*children)
 
 
 @jax.tree_util.register_pytree_node_class
@@ -412,12 +408,11 @@ class ElasticSqKOverlap(RegTICost):
   Args:
     k: Number of groups. Must be in ``[0, d)`` where :math:`d` is the
       dimensionality of the data.
-    gamma: Strength of the squared k-overlap norm regularization.
+    gamma: Strength of the squared k-overlap norm regularization, :math:`> 0`.
   """
 
   def __init__(self, k: int, gamma: float = 1.0):
     super().__init__()
-    assert gamma > 0, "Gamma must be positive."
     self.k = k
     self.gamma = gamma
 
@@ -488,12 +483,11 @@ def inner(r: int, l: int,
     return sgn * q[jnp.argsort(z_ixs.astype(float))]
 
   def tree_flatten(self):  # noqa: D102
-    return (), (self.k, self.gamma)
+    return (self.gamma,), {"k": self.k}
 
   @classmethod
   def tree_unflatten(cls, aux_data, children):  # noqa: D102
-    del children
-    return cls(*aux_data)
+    return cls(**aux_data, gamma=children[0])
 
 
 @jax.tree_util.register_pytree_node_class