On the volumetric path

We consider the logarithmic and the volumetric barrier functions used in interior point methods. In the case of the logarithmic barrier function, the anaytic center of a level set is the point at which the central path intersects that level set. We prove that this also holds for the volumetric path. For the central path, it is also true that the analytic center of the optimal level set is the limit point of the central path. The only known case where this last property with the logarithmic barrier function fails occurs in case of semidefinite optimization in the absence of strict complementarity. By an example we show that this property does not hold even for a linear optimization problem in canonical form for the volumetric path.