Renormings and symmetry properties of 1-greedy bases

We continue the study of 11-greedy bases initiated by Albiac and Wojtaszczyk (2006) [1]. We answer several open problems that they raised concerning symmetry properties of 11-greedy bases and the improving of the greedy constant by renorming. We show that 11-greedy bases need not be symmetric or subsymmetric. We also prove that one cannot in general make a greedy basis 11-greedy as demonstrated for the Haar basis of the dyadic Hardy space H1(R)H1(R) and for the unit vector basis of Tsirelson space. On the other hand, we give a renorming of LpLp (1