In this work, we investigate the question of how knowledge about expectations $\mathbb{E}(f_i(X))$ of a random vector $X$ translate into inequalities for $\mathbb{E}(g(X))$ for given functions $f_i$, $g$ and a random vector $X$ whose support is contained in some set $S\subseteq \mathbb{R}^n$. We show that there is a connection between the problem of obtaining tight expectation inequalities in this context and properties of convex hulls, allowing us to rewrite it as an optimization problem. The results of these optimization problems not only arrive at sharp bounds for $\mathbb{E}(g(X))$ but in some cases also yield discrete probability measures where equality holds. We develop an analytical approach that is particularly suited for studying the Jensen gap problem when the known information are the average and variance, as well as a numerical approach for the general case, that reduces the problem to a convex optimization; which in a sense extends known results about the moment problem.