Some results on centers of polytopes

The main ingredient for polynomiality in interior point methods is the centering procedure. All interior point algorithms for solving linear programing (LP) problems, known to be polynomial, have an explicit or implicit mechanism for finding a center of the LP polytope. Therefore, we consider the study of centers of polytopes as an important undertaking. In this work, we describe three notions of centers of a polytope and their relations. We show that if we can compute any one of them in polynomial time, then we can solve LP in polynomial time.20192