Barriers on projective convex sets
Résumé
Modern interior-point methods used for optimization on convex sets in a ne space are based on the notion of a barrier function. Projective space lacks crucial properties inherent to a ne space, and the concept of a barrier function cannot be directly carried over. We present a self-contained theory of barriers on convex sets in projective space which is build upon the projective cross-ratio. Such a projective barrier equips the set with a Codazzi structure, which is a generalization of the Hessian structure induced by a barrier in the a ne case. The results provide a new interpretation of the a ne theory and serve as a base for constructing a theory of interior-point methods for projective convex optimization.