Super-Gaussian directions of random vectors

We establish the following universality property in high dimensions: Let $X$ be a random vector with density in $\mathbb{R}^n$. The density function can be arbitrary. We show that there exists a fixed unit vector $\theta \in \mathbb{R}^n$ such that the random variable $Y = \langle X, \theta \rangle$ satisfies $$\min \left \{ \mathbb{P}( Y \geq t M ), \mathbb{P}(Y \leq -tM) \right \} \geq c e^{-C t^2} \qquad \qquad \text{for all} \ 0 \leq t \leq \tilde{c} \sqrt{n},$$ where $M > 0$ is any median of $|Y|$, i.e., $\min \{ \mathbb{P}( |Y| \geq M), \mathbb{P}( |Y| \leq M ) \} \geq 1/2$. Here, $c, \tilde{c}, C > 0$ are universal constants. The dependence on the dimension $n$ is optimal, up to universal constants, improving upon our previous work.