In $L^p((X, \mu), \mathbb{R})$, $p \geq 2$, it is possible to define the norm (considering that $\mu(X) = 1$.)
$$||f||_2 = \left( \int_X f(x)^2d \mu(x) \right)^{1/2}$$
which comes from the following inner product
$$<f, g> = \int_X f(x) g(x) d\mu(x) $$
It is also possible to define all p norms
$$||f||_p = \left( \int_X |f(x)|^p d\mu(x) \right)^{1/p}$$
It is possible to prove that none of these except the one for $p=2$ come from an inner product (correct me if I am wrong), and to generally prove that a given norm comes from an inner product, one needs to show that this norm verifies the parallelogram law.
The question is the following: Are there interesting known ways of constructing such norms that are guaranteed to come from an inner product of the form
$$||f|| = H\left( \int_X G(f(x)) d\mu(x) \right) $$
where $G: \mathbb{R} \mapsto \mathbb{R}$, $H: \mathbb{R} \mapsto \mathbb{R}$ and ignoring for now the question of whether $G(f(x))$ is actually integrable on $X$.
Let me know if this needs clarification.