Characterizing Centers of Convex Bodies via Optimization

General planar center points are defined via optimization theory as the minimizing solutions to particular sums or products of distance functions. A special class of centers using a logarithmic distance functional is then identified. A vector characterization of the points is obtained using elementary techniques. It is shown directly that points which solve similar minimization problems have similar characterizations. Standard algebraic and geometric arguments are used throughout the paper, and results are discussed in R2 and Rn, n≥3. This approach is a natural outgrowth of the recent literature on analytic centers and interior point techniques, and as outlined unifies many of the classic invariant points of polygonal systems.