Let's say that random variable $X$ is $\sigma$-subGaussian about a point $c \in \mathbb R$ if $\mathbb E[\Psi_2(\sigma |X-c|)] \le 1$, where $\Psi_2(t):=e^{t^2}-1$. Now, suppose the random variable $X_k$ is $\sigma$-SubGaussian about $c_k \in \mathbb R$, for $k=1,2$.
Question. . Find $\alpha \ge 0$ as small as possible such that $\mathbb E[\Psi_2(|X_1-c|)] + \mathbb E[\Psi_2(|X_2-c|)] \le \alpha$, for some $c \in \mathbb R$.