Find $\sup_y \mathbb{E}[e^{-\alpha \|X - y\|^2}]$

Question

Let $X: \Omega \to \mathcal{X}$ be a random variable that induces a probability measure $\mathbb{P}$ on $\mathcal{X}$. Let's say $\mathcal{X}$ is a compact subset of the Euclidean space. I want to find (or give a non-trivial upper bound for): $$ \sup_{y \in \mathcal{X}} \mathbb{E}[e^{-\alpha \|X - y\|^2}] = \sup_{y \in \mathcal{X}} \int e^{-\alpha \|x - y\|^2} d \mathbb{P}(x) $$ for any $\alpha \geq 0$. More generally, I'm trying to find $ \sup_{y \in \mathcal{Y}} \mathbb{E}[e^{-\alpha \rho(X, y)}] $ for a generic function $\rho: \mathcal{X} \times \mathcal{Y} \to [0, \infty)$. The seemingly related problem $\sup_y \mathbb{E}[-\|X - y\|^2] = - \inf_y \mathbb{E}[\|X - y\|^2]$ has a closed-form solution at $y^*=\mathbb{E}[X]$, so I'm hoping the more general problem also has a solution in terms of the moments of $X$.