Some isomorphically polyhedral Orlicz sequence spaces

A Banach space is polyhedral if the unit ball of each of its finite dimensional subspaces is a polyhedron. It is known that a polyhedral Banach space has a separable dual and is c0-saturated, i.e., each closed infinite dimensional subspace contains an isomorph of c0. In this paper, we show that the Orlicz sequence space hM is isomorphic to a polyhedral Banach space if limt→0 M(Kt)/M(t) = ∞ for some K < ∞. We also construct an Orlicz sequence space hM which is c0saturated, but which is not isomorphic to any polyhedral Banach space. This shows that being c0-saturated and having a separable dual are not sufficient for a Banach space to be isomorphic to a polyhedral Banach space. A Banach space is said to be polyhedral if the unit ball of each of its finite dimensional subspaces is a polyhedron. It is isomorphically polyhedral if it is isomorphic to a polyhedral Banach space. Fundamental results concerning polyhedral Banach spaces were obtained by Fonf [1, 2]. Theorem 1 (Fonf) A separable isomorphically polyhedral Banach space is c0-saturated and has a separable dual. Recall that a Banach space is c0-saturated if every closed infinite dimensional subspace contains an isomorph of c0. Fonf also proved a characterization of isomorphically polyhedral spaces in terms of certain norming subsets in the dual. In order to state the relevant results, we introduce some terminology due to Rosenthal [4, 5]. The (closed) unit ball of a Banach space E is denoted by UE. Definition Let E be a Banach space. (1) A subset W ⊆ E ′ is precisely norming (p.n.) if W ⊆ UE′, and for all x ∈ E, there is a w ∈ W such that ‖x‖ = |w(x)|. (2) A subset W ⊆ E ′ is isomorphically precisely norming (i.p.n.) if W is bounded and 1991 Mathematics Subject Classification 46B03, 46B20, 46B45.