ON TREE STRUCTURES IN BANACH SPACES

The purpose of this note is to relate the structure of flat Banach spaces to the "trees " of James. Based on this and some recent work of several other authors, we explore a similar relation for two further classes of spaces, namely spaces which are dual to spaces containing a subspace isomor-phic to h and spaces which are dual to spaces that are not separable. A Banach space X is said to be flat if the girth of its unit ball (defined by Sehaffer [14] to be the infimum of the lengths of all centrally symmetric curves which lie in the surface of the unit ball) is four and if the girth is achieved by some curve (i.e., the infimum is a minimum). This is equivalent to the statement that there exists a function g: [0, 2]-+X such that | | flr(ί) | | = 1 for each t e [0, 2] , g(0) =-#(2), and g is Lipschitz continuous with constant 1. Examples, consisting of common spaces, appear below. Some dis-tinguished geometric properties of flat Banach spaces, including the ones which give rise to the term flat, were given by the authors in [3] and [4]. A Banach space X is said to have the infinite tree property (James, [6]) if for some ε> 0, X contains a tree with an infinite number of branches, i.e., there are elements x((2ί — l)/2%), i = 1,-2n~\ n = l,2, •••, in the unit ball of X which branch at each x ( ) ac-cording to and —¥~) ~Λ ^ ) + 1 \ The space is said to have the finite tree property (James, [6] if for some e> 0 and for each positive integer N, X contains a tree with N branchings (i.e., n = 1, , N). At present it is known that X has the finite tree property if and only if X is not super-reflexive (James, [8]) if and only if the girth of the unit ball of X is equal to four (James and Schaffer, [9]). I