Given a basis in a Banach space $X$, can one find, for every $\varepsilon>0$, an equivalent basis with basis constant at most $1+\varepsilon$?
In $L_p[0,1]$ with $1<p<\infty$ any monotone basis is unconditional so one cannot expect better than $1+\varepsilon$ in general. What about $L_1[0,1]$, is every basis equivalent with a monotone basis?
Edit: The original question has a negative answer, thank you for the reference. What about the $L_p[0,1]$ case? Any known results in this direction?
P.Enflo (Arkiv Mat., 11 (1973), 103-107) constructed a Banach space $X$ with a basis such that the basis constant constant of any basis in $X$ is $\ge c_X>1$.
The answer is no. In the classical paper of Gurariy-Gurariy Bases in uniformly convex and uniformly smooth Banach spaces. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 35 1971 210–215. they proved that if we have a quasinormalised normalised basis $(b_j)$ with basis constant $C$ in a uniformly convex and uniformly smooth space then there exists $1<p\leq q<\infty$ and $A,B$ that depends only on C, quasisnormalisation and moduli of smoothness and convexity such that for $x=\sum_j a_j b_j$ we have $$A\|(a_j)\|_q\leq \|x\|\leq B\|(a_j)\|_p.$$ This was later done for super-reflexive spaces by James. It is known and easy that Babenko examples( Babenko On conjugate functions, Doklady AN SSSR 62(1948) 157-160 (in Russian)) of conditional bases in Hilbert spaces may need $p$ very close to $1$ and $q$ very big.
All this shows that given $C<\infty$ there exists a basis in a Hilbert space such that every basis in a Hilbert space equivalent to it has a basis constant $\geq C$.
