Renorming spaces with greedy bases

We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given e 0 , so that the basis becomes ( 1 + e ) -democratic, and hence ( 2 + e ) -greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is ( 1 + e ) -greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applications include the Haar systems in L p 0 , 1 , 1 < p < ∞ , and in dyadic Hardy space H 1 , as well as the unit vector basis of Tsirelson space.