A Locally Well-behaved Potential Function and a Simple Newton-type Method for Finding the center of a Polytope

We show that the first two terms in the power series of F (x) at x sub 0 serve as a good approximation to F(x) in a suitable ellipsoid around x sub 0, and that minimizing the first order (linear) term in the power series over this ellipsoid increases F (x) by a fixed additive constant as long as x sub 0 is not too close to the center omega.