Let $d$ be a large positive integer. Fix a unit-vector $v \in \mathbb R^d$, and scalars $b,c \in \mathbb R$ with $b > 0$. Let $X$ be a log-concave random vector in $\mathbb R^d$ normalized so that $\mathbb E[(1/d)\|X\|^2] = 1$, WLOG.

Question 1. Is there a nontrivial lower-bound for $\alpha:=\mathbb E[e^{-b(v^\top X - c)^2}]$ in terms of $b$ and $c$ ?

For example, if $X$ is distributed according to $N(0,I_d)$, then $Z:=v^\top X$ has distribution $N(0,1)$, and so direct integration gives $$ \alpha = \mathbb E_{Z \sim N(0,1)}[e^{-b(Z-c)^2}] = \sqrt{\frac{b}{1 + 2b}}e^{-bc^2/(1 + 2b)}. $$

Question 2. Same question as Question 1, additional condition that $X$ is isotropic.

Let $d$ be a large positive integer. Fix a unit-vector $v \in \mathbb R^d$, and scalars $b,c \in \mathbb R$ with $b > 0$. Let $X$ be a log-concave random vector in $\mathbb R^d$ normalized so that $\mathbb E[(1/d)\|X\|^2] = 1$, WLOG.

Question 1. Is there a nontrivial lower-bound for $\alpha:=\mathbb E[e^{-b(v^\top X - c)^2}]$ in terms of $b$ and $c$ ?

For example, if $X$ is distributed according to $N(0,I_d)$, then $Z:=v^\top X$ has distribution $N(0,1)$, and so direct integration gives $$ \alpha = \mathbb E_{Z \sim N(0,1)}[e^{-b(Z-c)^2}] = \sqrt{\frac{1}{1 + 2b}}e^{-bc^2/(1 + 2b)}. $$

Question 2. Same question as Question 1, additional condition that $X$ is isotropic.

Let $d$ be a large positive integer. Fix a unit-vector $v \in \mathbb R^d$, and scalars $b,c \in \mathbb R$ with $b > 0$. Let $X$ be a log-concave random vector in $\mathbb R^d$ normalized so that $\mathbb E[(1/d)\|X\|^2] = 1$, WLOG.

Question 1. Is there a nontrivial lower-bound for $\alpha:=\mathbb E[e^{-b(v^\top X - c)^2}]$ ?

For example, if $X$ is distributed according to $N(0,I_d)$, then $Z:=v^\top X$ has distribution $N(0,1)$, and so $$ \alpha = \mathbb E_{Z \sim N(0,1)}[e^{-b(Z-c)^2}] = \sqrt{\frac{b}{1 + 2b}}e^{-bc^2/(1 + 2b)}. $$

Question 2. Same question as Question 1, additional condition that $X$ is isotropic.

Let $d$ be a large positive integer. Fix a unit-vector $v \in \mathbb R^d$, and scalars $b,c \in \mathbb R$ with $b > 0$. Let $X$ be a log-concave random vector in $\mathbb R^d$ normalized so that $\mathbb E[(1/d)\|X\|^2] = 1$, WLOG.

Question 1. Is there a nontrivial lower-bound for $\alpha:=\mathbb E[e^{-b(v^\top X - c)^2}]$ in terms of $b$ and $c$ ?

For example, if $X$ is distributed according to $N(0,I_d)$, then $Z:=v^\top X$ has distribution $N(0,1)$, and so direct integration gives $$ \alpha = \mathbb E_{Z \sim N(0,1)}[e^{-b(Z-c)^2}] = \sqrt{\frac{b}{1 + 2b}}e^{-bc^2/(1 + 2b)}. $$

Question 2. Same question as Question 1, additional condition that $X$ is isotropic.