  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\to X such that P(B(Y)) is a subset of A (by B(X) we
denote the unit ball).
  The notion of sufficient enlargement is implicit in the paper: B.Grunbaum,
Projection constants, Trans. Amer. Math. Soc. 95 (1960) 451--465. It was
explicilty introduced by the author in: M.I.Ostrovskii, Generalization of
projection constants: sufficient enlargements, Extracta Math., 11 (1996),
466--474.
  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.
