On trees and dual rotund norms

We give a necessary and sufficient condition for the existence of an equivalent dual rotund norm on C0( )∗ ≡ 1( ), where is a tree. The condition is expressed succinctly, in terms of the embeddability of 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. © 2005 Elsevier Inc. All rights reserved.