I have been struggling with the following question.
Let $X \in \mathbb{R}^n$ be a $K$ sub Gaussian random vector (i.e. $\|\langle u, X \rangle\|_{\psi_2} \leq K$ for all $\|u\|=1$) and let $f : \mathbb{R}^n \to \mathbb{R}$ be a $1$-Lipschitz function. Is it possible to show that $f(X)$ is sub Gaussian with constant independent of $n$ ?
A similar question was asked here and here, in particular, it was mentioned that the previous property holds when the vector is a standard Gaussian (see for instance Theorem 8 here). An answer proposed to see $X$ as $\phi(Z)$ with some Lipschitz function $\phi$ and Gaussian $Z$. Another situation where this can be proved is if $X$ has density $e^{-U(x)}$ for strongly convex $U$ (Theorem 5.2.15 in the book High Dimensional Probability by Roman Vershynin). Unfortunately I cannot leverage these options.
It is important to note this is untrue when $X$ is formed of independent separately sub Gaussian coordinates (see this thread) but that is a different setting.