On non-midpoint locally uniformly rotund normability in Banach spaces

We will show that if X is a tree-complete subspace of ∞ , which contains c 0 , then it does not admit any weakly midpoint locally uniformly convex renorming. It follows that such a space has no equivalent Kadec renorming. 1. Introduction. It is known that ∞ has an equivalent strictly convex renorming [2]; however, by a result due to Lindenstrauss, it cannot be equivalently renormed in locally uniformly convex manner [10]. In this note, we will show that every tree-complete subspace of ∞ , which contains c 0 , does not admit any equivalent weakly midpoint locally uniformly convex norm. This can be considered as an extension of [1, 8]. Since every strictly convexifiable Banach space with Kadec property admits an equivalent midpoint locally uniformly convex renorming [9], it follows that every subspace of ∞ with the tree-completeness property has no equivalent Kadec renorming. The existence of such a (nontrivial) subspace, which does not contain any copy of ∞ , has already been proved by Haydon and Zizler (see [5, 7]).