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I did quite a few numerical computations and think the following is true, but I cannot prove it:

Let $\varphi(x):=\sum_{i=1}^n \varphi_i(x_i)$ where $x=(x_1,...,x_n) \in \mathbb{R}^n$ and $\varphi_i$ are even scalar convex functions such that $\varphi''$ is strictly increasing on $[0,\infty).$

We then define a probability measure (which under apppropriate normalization) is defined as $$p_y(x) \propto e^{\langle y, x\rangle}e^{-\varphi(x) } \ dx. $$

Can we show that for all unit vectors $z \in \mathbb{R}^n$ we have for all $y \in \mathbb{R}^n$

$$Var_{p_0}(\langle z,X_0 \rangle_{\mathbb{R}^n}) \ge Var_{p_y}(\langle z,X_y \rangle_{\mathbb{R}^n})?$$

In other words, the variance of $\langle z,X_y\rangle$ where $X_y$ is distributed according to $p_y$ is maximized at $y=0$ for any unit vector $z.$

Is this a known theorem or somehow easy to show?-Any pointers are highly appreciated.

I did quite a few numerical computations and think the following is true, but I cannot prove it:

Let $\varphi(x):=\sum_{i=1}^n \varphi_i(x_i)$ where $x=(x_1,...,x_n) \in \mathbb{R}^n$ and $\varphi_i \in C^{\infty}$ are even scalar convex functions such that $\varphi''$ is strictly increasing on $[0,\infty).$

We then define a probability measure (which under apppropriate normalization) is defined as $$p_y(x) \propto e^{\langle y, x\rangle}e^{-\varphi(x) } \ dx. $$

Can we show that for all unit vectors $z \in \mathbb{R}^n$ we have for all $y \in \mathbb{R}^n$

$$Var_{p_0}(\langle z,X_0 \rangle_{\mathbb{R}^n}) \ge Var_{p_y}(\langle z,X_y \rangle_{\mathbb{R}^n})?$$

In other words, the variance of $\langle z,X_y\rangle$ where $X_y$ is distributed according to $p_y$ is maximized at $y=0$ for any unit vector $z.$

Is this a known theorem or somehow easy to show?-Any pointers are highly appreciated and please let me know if there are any questions.

I did quite a few numerical computations and think the following is true, but I cannot prove it:

Let $\varphi(x):=\sum_{i=1}^n \varphi_i(x_i)$ where $x=(x_1,...,x_n) \in \mathbb{R}^n$ and $\varphi_i$ are even scalar convex functions such that $\varphi''$ is strictly increasing on $[0,\infty).$

We then define a probability measure (which under apppropriate normalization) is defined as $$p_y(x) \propto e^{\langle y, x\rangle}e^{-\varphi(x) } \ dx. $$

Can we show that for all unit vectors $z \in \mathbb{R}^n$ we have for all $y \in \mathbb{R}^n$

$$Var_{p_0}(\langle z,X_0 \rangle_{\mathbb{R}^n}) \ge Var_{p_y}(\langle z,X_y \rangle_{\mathbb{R}^n})?$$

In other words, the variance of $\langle z,X_y\rangle$ where $X_y$ is distributed according to $p_y$ is maximized at $y=0$ for any unit vector $z.$

Is this a known theorem or somehow easy to show?

I did quite a few numerical computations and think the following is true, but I cannot prove it:

Let $\varphi(x):=\sum_{i=1}^n \varphi_i(x_i)$ where $x=(x_1,...,x_n) \in \mathbb{R}^n$ and $\varphi_i$ are even scalar convex functions such that $\varphi''$ is strictly increasing on $[0,\infty).$

We then define a probability measure (which under apppropriate normalization) is defined as $$p_y(x) \propto e^{\langle y, x\rangle}e^{-\varphi(x) } \ dx. $$

Can we show that for all unit vectors $z \in \mathbb{R}^n$ we have for all $y \in \mathbb{R}^n$

$$Var_{p_0}(\langle z,X_0 \rangle_{\mathbb{R}^n}) \ge Var_{p_y}(\langle z,X_y \rangle_{\mathbb{R}^n})?$$

In other words, the variance of $\langle z,X_y\rangle$ where $X_y$ is distributed according to $p_y$ is maximized at $y=0$ for any unit vector $z.$

Is this a known theorem or somehow easy to show?-Any pointers are highly appreciated.

I did quite a few numerical computations and think the following is true, but I cannot prove it:

Let $\varphi(x):=\sum_{i=1}^n \varphi_i(x_i)$ where $x=(x_1,...,x_n) \in \mathbb{R}^n$ and $\varphi_i$ are even scalar convex functions such that $\varphi''$ is strictly increasing on $[0,\infty).$

We then define a probability measure (which under apppropriate normalization) is defined as $$p_y(x) \propto e^{\langle y, x\rangle}e^{-\varphi(x) } \ dx. $$

Can we show that for all unit vectors $z \in \mathbb{R}^n$ we have for all $y \in \mathbb{R}^n$

$$Var_{p_0}(\langle z,X_0 \rangle_{\mathbb{R}^n}) \ge Var_{p_y}(\langle z,X_y \rangle_{\mathbb{R}^n})?$$

In other words, the variance of $\langle z,X_y\rangle$ where $X_y$ is distributed according to $p_y$ is maximized at $y=0$ for any unit vector $z.$

Is this a known theorem or somehow easy to show?

Please let me know if you have any questions and see the interesting remarks in the response/comment below.

I did quite a few numerical computations and think the following is true, but I cannot prove it:

Let $\varphi(x):=\sum_{i=1}^n \varphi_i(x_i)$ where $x=(x_1,...,x_n) \in \mathbb{R}^n$ and $\varphi_i$ are even scalar convex functions such that $\varphi''$ is strictly increasing on $[0,\infty).$

We then define a probability measure (which under apppropriate normalization) is defined as $$p_y(x) \propto e^{\langle y, x\rangle}e^{-\varphi(x) } \ dx. $$

Can we show that for all unit vectors $z \in \mathbb{R}^n$ we have for all $y \in \mathbb{R}^n$

$$Var_{p_0}(\langle z,X_0 \rangle_{\mathbb{R}^n}) \ge Var_{p_y}(\langle z,X_y \rangle_{\mathbb{R}^n})?$$

In other words, the variance of $\langle z,X_y\rangle$ where $X_y$ is distributed according to $p_y$ is maximized at $y=0$ for any unit vector $z.$

Is this a known theorem or somehow easy to show?