The relations between nonlinear projections and the geometric properties of a Banach space

We call infinite dimensional spaces equipped with the complete topology which is induced by "norm" Banach spaces. For example, the set of all continuously differentiable functions, the set of all integrable functions and so on. A Banach space plays an important role in a probability theory, an optimizing theory and so on.
I study the geometric properties of Banach spaces by using nonlinear projections of the whole space. Nonexpansve projections and metric projections are well known projections and recently generalized projections and generalized nonexpansive projections are discovered. I am trying to find the relations between these nonlinear projections and linear projections, for example conditional expectations of L^p functions, by using the orthogonal decomposition of a Banach space.