It could be as simple as calling from a shell in your app:

kubectl create namespace <your-namespace-name>

Essentially, kubectl talks to the kube-apiserver.

You can also directly call the kube-apiserver. This is an example to list the pods:

$ curl -k -H 'Authorization: Bearer <token>' \
              https://$KUBERNETES_SERVICE_HOST:6443/api/<api-version>/namespaces/default/pods

More specifically to create a namespace:

$ curl -k -H -X POST -H 'Content-Type: application/json' \
                     -H 'Authorization: Bearer <token>' \
                     https://$KUBERNETES_SERVICE_HOST:6443/api/v1/namespaces/ -d '
{
    "apiVersion": "v1",
    "kind": "Namespace",
    "metadata": {
        "name": "mynewnamespace"
    }
}'

In case you are wondering about the <token>, it's a Kubernetes Secret typically belonging to a ServiceAccount and bound to a ClusterRole that allows you to create namespaces.

You can create a Service Account like this:

$ kubectl create serviceaccount namespace-creator

Then you'll see the token like this (a token is automatically generated):

$ kubectl describe sa namespace-creator
Name:                namespace-creator
Namespace:           default
Labels:              <none>
Annotations:         <none>
Image pull secrets:  <none>
Mountable secrets:   namespace-creator-token-xxxxx
Tokens:              namespace-creator-token-xxxxx
Events:              <none>

Then you would get the secret:

$ kubectl describe secret namespace-creator-token-xxxxx
Name:         namespace-creator-token-xxxx
Namespace:    default
Labels:       <none>
Annotations:  kubernetes.io/service-account.name: namespace-creator
              kubernetes.io/service-account.uid: <redacted>

Type:  kubernetes.io/service-account-token

Data
====
ca.crt:     1025 bytes
namespace:  7 bytes
token:      <REDACTED> <== This is the token you need for Authorization: Bearer

Your ClusterRole should look something like this:

kind: ClusterRole
apiVersion: rbac.authorization.k8s.io/v1
metadata:
  name: namespace-creator
rules:
- apiGroups: ["*"]
  resources: ["namespaces"]
  verbs: ["create"]

Then you would bind it like this:

$ kubectl create clusterrolebinding namespace-creator-binding --clusterrole=namespace-creator --serviceaccount=namespace-creator

When it comes to writing code you can use any HTTP client library in any language to call the same endpoints.

There are also libraries like the client-go library that takes care of the plumbing of connecting to a kube-apiserver.