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 $\text{vare}>0$, so that the basis becomes $(1+\text{vare})$-democratic, and hence $(2+\text{vare})$-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 $(1+\text{vare})$-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 $L_p[0,1]$, $1<p<\mathrm{\infty }$, and in dyadic Hardy space $H_1$, as well as the unit vector basis of Tsirelson space.