A Cheney and Wulbert type lifting theorem in optimization

In this paper we exhibit new classes of Banach spaces for which strong notions of optimization can be lifted from quotient spaces. Motivated by a well known result of Cheney and Wulbert on lifting of proximinality from a quotient space to a subspace, for closed subspaces, Z ⊂ Y ⊂ X , we consider stronger forms of optimization, that Z has in X and the quotient space Y / Z has in X / Z should lead to the conclusion Y has the same property in X . The versions we consider have been studied under various names in the literature as L -proximinal subspaces or subspaces that have the strong- 1 1 2 -ball property. We give an example where the strong- 1 1 2 -ball property fails to lift to the quotient. We show that if every M -ideal in Y is a M -summand, for a finite codimensional subspace Z ⊂ Y , that is a M -ideal in X with the strong- 1 1 2 -ball property in X and if Y / Z has the 1 1 2 -ball property in X / Z , then Y has the strong- 1 1 2 -ball property in X .