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 $vare>0$, so that the basis becomes $(1+vare)
$-democratic, and hence $(2+vare)$-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 $(1+vare)$-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 $L_p[0,1]$, $1<p<infty$, and in dyadic Hardy space $H_1$, as well as the unit vector basis of Tsirelson space.
Let $X$ be an infinite dimensional uniformly smooth Banach space. We prove that $X$ contains an infinite equilateral set. That is, there exists a constant $lambda>0$ and an infinite sequence $(x_i)_{i=1}^inftysubset X$ such that $|x_i-x_j|=lambda$ for all $i eq j$.
We prove thatthe Banach space $(oplus_{n=1}^infty ell_p^n)_{ell_q}$, which is isomorphic to certain Besov spaces, has a greedy basis whenever $1leq p leqinfty$ and $1<q<infty$. Furthermore, the Banach spaces $(oplus_{n=1}^infty ell_p^n)_{ell_1}$, wit
h $1<ple infty$, and $(oplus_{n=1}^infty ell_p^n)_{c_0}$, with $1le p<infty$ do not have a greedy bases. We prove as well that the space $(oplus_{n=1}^infty ell_p^n)_{ell_q}$ has a 1-greedy basis if and only if $1leq p=qle infty$.
For every $alpha<omega_1$ we establish the existence of a separable Banach space whose Szlenk index is $omega^{alphaomega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed $omega^{alphaomega}$. In order to
prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with upper estimates.
