Let $X_1,\dots,X_n$ be $n$ i.i.d random variables taking values in a Polish vector space $\mathcal{X}$ and with (Borel) probability distribution $\mu$.

For any convex, compact $\Gamma \subset \mathcal{X}$, it can be proved using Sion's minimax theorem (see for example Exercice 4.5.5 in Dembo and Zeitouni) that: $$ \forall n \quad \mathbb{P}\left(\frac{1}{n}\sum_{k=1}^{n}{X_k} \in \Gamma\right) \leq e^{-n \inf_{x\in \Gamma}{I(x)}}$$ with $I$ the Fenchel Legendre transform of the moment generating function.

Using the concept of dominating point (see for example the works of Ney), it can also be shown that for any convex, open set $\Gamma$ (and a few more assumptions): $$ \forall n \quad \mathbb{P}\left(\frac{1}{n}\sum_{k=1}^{n}{X_k} \in \Gamma\right) \leq e^{-n \inf_{x\in \Gamma}{I(x)}}$$ The additional assumptions are actually needed to ensure the existence of a dominating point which implies a much stronger result than just the inequality. It is not obvious to me whether the additional assumptions are needed if we're just interested in this inequality?

Actually, let me ask a more general question: are there examples of convex (measurable) $\Gamma$ such that: $$ \forall n \quad \mathbb{P}\left(\frac{1}{n}\sum_{k=1}^{n}{X_k} \in \Gamma\right) \leq e^{-n \inf_{x\in \Gamma}{I(x)}}$$ doesn't hold?

Thank you for your insights.