Renormings and symmetry properties of 1-greedy bases

We continue the study of 1-greedy bases initiated by Albiac and Wojtaszczyk (2006) [1]. We answer several open problems that they raised concerning symmetry properties of 1-greedy bases and the improving of the greedy constant by renorming. We show that 1-greedy bases need not be symmetric or subsymmetric. We also prove that one cannot in general make a greedy basis 1-greedy as demonstrated for the Haar basis of the dyadic Hardy space H(1) (R) and for the unit vector basis of Tsirelson space. On the other hand, we give a renorming of L(p) (1 < p < infinity) that makes the Haar basis 1-unconditional and 1-democratic. Other results in this paper clarify the relationship between various basis constants that arise in the context of greedy bases. (C) 2011 Elsevier Inc. All rights reserved.