Is the computational complexity of finding or approximating $\inf\{c^Tx:x\in X\}$ (where $X$ is a compact convex given explicitly or by some reasonable oracle) known?
EDIT: Suppose we had an efficient separation oracle. Would I be correct in saying that for $X \subseteq \mathbb{R}^n$, the runtime would be $\tilde O(n^6\log n)$, where the tilde notation suppresses log-dependence on the data, and also suppresses the runtime of the oracle?