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?



Basis equivalent with a monotone basis
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 $\ge c\_X>1$.


