Let $1<p<\infty$. Recall that a Banach space $X$ is $l\_{p}$-saturated if every infinite-dimensional subspace of $X$ contains a subspace isomorphic to $l\_{p}$. I have a seemingly stronger notion of $l\_{p}$-saturated Banach spaces. We say that a Banach space $X$ is strongly $l\_{p}$-saturated if for every infinite-dimensional subspace $M$ and every $\epsilon>0$, there exists an infinite-dimensional subspace $N$ of $M$ such that $N$ is $(1+\epsilon)$-isomorphic to $l\_{p}$. I have the following two questions:


Question 1. If a Banach space $X$ is $l\_{p}$-saturated, is $X$ strongly $l\_{p}$-saturated?


Question 2. Are there other examples of strongly $l\_{p}$-saturated spaces besides $l\_{p}$? Is $L\_{p}$ strongly $l\_{p}$-saturated?


Thank you!



Two questions on $l_{p}$-saturated Banach spaces
As for question 1; this is the famous [distortion problem](https://en.wikipedia.org/wiki/Distortion_problem). Your question has negative answer already for some renorming of $\ell\_p$:



> 
> E. Odell, Th. Schlumprecht, The distortion problem of Hilbert space, *Geom. Funct. Anal.*, **3**, 201–207.
> 
> 
> 


As for your second question, $L\_p$ is not $\ell\_p$-saturaded as the Rademachers span a copy of $\ell\_2$.


