A characterization and equations for minimal shape-preserving projections

Let X denote a (real) Banach space and V an n-dimensional subspace. We denote by B=B(X,V) the space of all bounded linear operators from X into V; let P(X,V) be the set of all projections in B. For a given , we denote by P=PS(X,V) the set of operators P∈P such that PS⊂S. When PS≠∅, we characterize those P∈PS for which ∥P∥ is minimal. This characterization is then utilized in several applications and examples.