2 00 2 Projections in Normed Linear Spaces and Sufficient Enlargements

Definition. A symmetric with respect to 0 bounded closed convex set A in a finite dimensional normed space X is called a sufficient enlargement for X (or of B(X)) if for arbitrary isometric embedding of X into a Banach space Y there exists a projection P : Y → X such that P (B(Y)) ⊂ A (by B we denote the unit ball). The main purpose of the present paper is to continue investigation of sufficient enlargements started in the papers cited above. In particular the author investigate sufficient enlargements whose support functions are in some directions close to those of the unit ball of the space, sufficient enlargements of minimal volume, sufficient enlargements for euclidean spaces. We denote the unit ball (sphere) of a normed linear space X by B(X) (S(X)). Convention. We shall use the term ball for symmetric with respect to 0 bounded closed convex set with nonempty interior in a finite dimensional linear space. Definition 1. A ball A in a finite dimensional normed space X is called a sufficient enlargement for X (or of B(X)) if for arbitrary isometric embedding X ⊂ Y (Y is a Banach space) there exists a projection P : Y → X such that P (B(Y)) ⊂ A. A minimal sufficient enlargement is defined to be a sufficient enlargement no proper subset of which is a sufficient enlargement. The notion of sufficient enlargement is implicit in B.Grünbaum's paper [2], it was explicilty introduced by the present author in [5]. The notion of sufficient enlargement is of interest because it is a natural geometric notion, it characterizes possible shadows of symmetric convex body onto a subspace, whose intersection with the body is given.