On trees and dual rotund norms

Abstract

We give a necessary and sufficient condition for the existence of an equivalent dual rotund norm on C 0 ( ϒ ) * ≡ ℓ 1 ( ϒ ) , where ϒ is a tree. The condition is expressed succinctly, in terms of the embeddability of ϒ into a particular totally ordered set, and compares very well with the analogous situation for local uniform rotundity. This resolves an open problem from Haydon's work in Asplund spaces, trees and renorming theory.