On well definedness of the central path

We study the well definedness of the central path for a linearly constraine d convex programming problem with smooth objective function. We prove that, under standard assumptions, existence of the central path is equivalent to the nonemptiness and boundedness of the optimal set. Other equivalent cond itions are given. We show that, under an additional assumption on the objec tive function, the central path converges to the analytic center of the opt imal set.