Let $f \colon \mathbb R^N \to \mathbb R$ be a smooth function. Let $\mu$ be a probability measure on $[0,1]$ and $X_1, \ldots , X_N$ be i.i.d. random variables on $\mathbb R$.

> **Question.** What is the maximum value of the expectation 
$$
\mathbb E[\vert f(X_1, \ldots , X_N) \vert] = \int_{\mathbb [0,1]^N} \vert f(x_1, \ldots , x_N) \vert  d\mu(x_1) \ldots d\mu(x_N)
$$
among all probability measures $\mu$ on $[0,1]$? 

This question arises from [this post][2] and from [this recent answer][3] of Sangchul Lee which seem however tailored for specific functions $f$ and particularly for the case $N=2$. 

I am very interested in the case $N\ge 3$; the function $f$ maybe be as smooth as needed (e.g. a polynomial). I have troubles in extending the (very elegant) variational approach of Sangchul Lee's to more variables, as no "bilinear form" is available. 

If I am not mistaken, if one considers only a.c. measures then the question becomes: maximize 
$$
\int_{[0,1]^N} |f(x_1, \ldots, x_N)| g(x_1) g(x_2) \ldots g(x_N) dx_1 dx_2 \ldots dx_N,
$$
among functions $g \ge 0$ such that $\int_0^1 g(s)\, ds = 1$. Can this be handled by the [general Holder's inequality][4]?

  [2]: https://math.stackexchange.com/questions/2209895/expected-absolute-difference-between-two-iid-variables/2542224#2542224
  [3]: https://math.stackexchange.com/questions/3112459/apex-angle-of-a-triangle-as-a-random-variable/3112772#3112772
  [4]: https://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality#Generalization_of_H%C3%B6lder's_inequality