In nearly all convex optimization methods that I read about, it is assumed that the problem satisfies Slater's condition, that is, there is a point that strictly satisfies all constraints (the interior of the feasible region is non-empty).

What if Slater's condition does not hold - is there a proof that, in this case, convex optimization cannot be solved in polynomial time unless P=NP? In other words: is Slater's condition a necessary condition for solving convex optimization problems in polynomial time?