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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 1. 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 and from this recent answer 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.

After Nate Eldredge's comment, it is not restrictive to consider only a.c. measures. Furthermore, we can initially consider a somehow easier case, where $f$ is a polynomial (then maybe work by uniform approximation). In addition, thanks to a clever "random thought" due to Pierre PC (see comments below), we can start assuming symmetry of the integrand. All in all, the problem "boils down" to the following:

Question 1 (easier version): Maximize $$ \int_{[0,1]^N} p(x_1, \ldots, x_N) g(x_1) g(x_2) \ldots g(x_N) dx_1 dx_2 \ldots dx_N, $$ among functions $g \ge 0$ such that $\int_0^1 g(s)\, ds = 1$, being $p \in \mathbb R[x_1, \ldots , x_N]$ a symmetric polynomial (of constant sign on [0,1]?).

Can this be handled by the general Holder's inequality?

Since the general case seems too hard, let me propose a toy model we can begin with. To me, it seems the difficult point is not the integrand (in view of the comment above general reduction arguments should lead to a very simple form of it) but rather the fact that $N \ge 3$ (in comparison with S. Lee's approach, which is tailored for $N=2$).

So here we go:

Question 2. (baby version) What is the maximum value of the expectation $$ \mathbb E[(X-Y)(Y-Z)(Z-X)] = \int_{\mathbb [0,1]^3} (X-Y)(Y-Z)(Z-X) d\mu(X) d\mu(Y)d\mu(Z) $$ among all probability measures $\mu$ on $[0,1]$?

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 1. 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 and from this recent answer 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.

After Nate Eldredge's comment, it is not restrictive to consider only a.c. measures. Furthermore, we can initially consider a somehow easier case, where $f$ is a polynomial (then maybe work by uniform approximation). In addition, thanks to a clever "random thought" due to Pierre PC (see comments below), we can start assuming symmetry of the integrand. All in all, the problem "boils down" to the following:

Question 1 (easier version): Maximize $$ \int_{[0,1]^N} p(x_1, \ldots, x_N) g(x_1) g(x_2) \ldots g(x_N) dx_1 dx_2 \ldots dx_N, $$ among functions $g \ge 0$ such that $\int_0^1 g(s)\, ds = 1$, being $p \in \mathbb R[x_1, \ldots , x_N]$ a symmetric polynomial (of constant sign on [0,1]?).

Can this be handled by the general Holder's inequality?

Since the general case seems too hard, let me propose a toy model we can begin with. To me, it seems the difficult point is not the integrand (in view of the comment above general reduction arguments should lead to a very simple form of it) but rather the fact that $N \ge 3$ (in comparison with S. Lee's approach, which is tailored for $N=2$).

So here we go:

Question 2. (baby version) What is the maximum value of the expectation $$ \mathbb E[|(X-Y)(Y-Z)(Z-X)|] = \int_{[0,1]^3} |(X-Y)(Y-Z)(Z-X)| d\mu(X) d\mu(Y)d\mu(Z) $$ among all probability measures $\mu$ on $[0,1]$?

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 1. 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 and from this recent answer 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.

After Nate Eldredge's comment, it is not restrictive to consider only a.c. measures. Furthermore, we can initially consider a somehow easier case, where $f$ is a polynomial (then maybe work by uniform approximation). In addition, thanks to a clever "random thought" due to Pierre PC (see comments below), we can start assuming symmetry of the integrand. All in all, the problem "boils down" to the following:

Question 1 (easier version): Maximize $$ \int_{[0,1]^N} p(x_1, \ldots, x_N) g(x_1) g(x_2) \ldots g(x_N) dx_1 dx_2 \ldots dx_N, $$ among functions $g \ge 0$ such that $\int_0^1 g(s)\, ds = 1$, being $p \in \mathbb R[x_1, \ldots , x_N]$ a symmetric polynomial (of constant sign on [0,1]?).

Can this be handled by the general Holder's inequality?

Since the general case seems too hard, let me propose a toy model we can begin with. To me, it seems the difficult point is not the integrand (in view of the comment above general reduction arguments should lead to a very simple form of it) but rather the fact that $N \ge 3$ (in comparison with S. Lee's approach, which is tailored for $N=2$).

So here we go:

Question 2. (baby version) What is the maximum value of the expectation $$ \mathbb E[XYZ] = \int_{\mathbb [0,1]^3} XYZ d\mu(X) d\mu(Y)d\mu(Z) $$ among all probability measures $\mu$ on $[0,1]$?

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 1. 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 and from this recent answer 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.

After Nate Eldredge's comment, it is not restrictive to consider only a.c. measures. Furthermore, we can initially consider a somehow easier case, where $f$ is a polynomial (then maybe work by uniform approximation). In addition, thanks to a clever "random thought" due to Pierre PC (see comments below), we can start assuming symmetry of the integrand. All in all, the problem "boils down" to the following:

Question 1 (easier version): Maximize $$ \int_{[0,1]^N} p(x_1, \ldots, x_N) g(x_1) g(x_2) \ldots g(x_N) dx_1 dx_2 \ldots dx_N, $$ among functions $g \ge 0$ such that $\int_0^1 g(s)\, ds = 1$, being $p \in \mathbb R[x_1, \ldots , x_N]$ a symmetric polynomial (of constant sign on [0,1]?).

Can this be handled by the general Holder's inequality?

Since the general case seems too hard, let me propose a toy model we can begin with. To me, it seems the difficult point is not the integrand (in view of the comment above general reduction arguments should lead to a very simple form of it) but rather the fact that $N \ge 3$ (in comparison with S. Lee's approach, which is tailored for $N=2$).

So here we go:

Question 2. (baby version) What is the maximum value of the expectation $$ \mathbb E[(X-Y)(Y-Z)(Z-X)] = \int_{\mathbb [0,1]^3} (X-Y)(Y-Z)(Z-X) d\mu(X) d\mu(Y)d\mu(Z) $$ among all probability measures $\mu$ on $[0,1]$?

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 1. 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 and from this recent answer 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.

After Nate Eldredge's comment, it is not restrictive to consider only a.c. measures. Furthermore, we can initially consider a somehow easier case, where $f$ is a polynomial (then maybe work by uniform approximation). All in all, the problem "boils down" to the following:

Question 1 (easier version): Maximize $$ \int_{[0,1]^N} p(x_1, \ldots, x_N) g(x_1) g(x_2) \ldots g(x_N) dx_1 dx_2 \ldots dx_N, $$ among functions $g \ge 0$ such that $\int_0^1 g(s)\, ds = 1$, being $p \in \mathbb R[x_1, \ldots , x_N]$ a polynomial (of constant sign on [0,1]?).

Can this be handled by the general Holder's inequality?

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 1. 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 and from this recent answer 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.

After Nate Eldredge's comment, it is not restrictive to consider only a.c. measures. Furthermore, we can initially consider a somehow easier case, where $f$ is a polynomial (then maybe work by uniform approximation). In addition, thanks to a clever "random thought" due to Pierre PC (see comments below), we can start assuming symmetry of the integrand. All in all, the problem "boils down" to the following:

Question 1 (easier version): Maximize $$ \int_{[0,1]^N} p(x_1, \ldots, x_N) g(x_1) g(x_2) \ldots g(x_N) dx_1 dx_2 \ldots dx_N, $$ among functions $g \ge 0$ such that $\int_0^1 g(s)\, ds = 1$, being $p \in \mathbb R[x_1, \ldots , x_N]$ a symmetric polynomial (of constant sign on [0,1]?).

Can this be handled by the general Holder's inequality?

Since the general case seems too hard, let me propose a toy model we can begin with. To me, it seems the difficult point is not the integrand (in view of the comment above general reduction arguments should lead to a very simple form of it) but rather the fact that $N \ge 3$ (in comparison with S. Lee's approach, which is tailored for $N=2$).

So here we go:

Question 2. (baby version) What is the maximum value of the expectation $$ \mathbb E[XYZ] = \int_{\mathbb [0,1]^3} XYZ d\mu(X) d\mu(Y)d\mu(Z) $$ among all probability measures $\mu$ on $[0,1]$?