Let  $A\subset \ell^2$  consist of  all  $x\in \ell^2$  with $|x|_2=1$ which  does not  belong to any $\ell^p$  for  all  $0<p<2$.

Note that $A$ is  non-empty with a  Baire category  argument.

I  am  interested in (somewhat) distribution and configuration of  $A$ and its complement in the  unit  sphere $S$ of  $\ell^2$.

Note that $S\setminus A$ is  path connected.

>Is $S\setminus A$ a  contractible  space?

> What can  be  said about connected  components of $A$? In particular  can one  find at least one  non constant  curve?


**Note:**  I was  thinking to a possible  density of  $A$ in the  sphere, or to a possible  measure of  $A$ as  a  subset of sphere. Note  that  density  of a  subset $A$ at a point $p$ of  a  metric  space  with measure $\mu$ is the  limit $\frac{\mu(A\cap B_\epsilon)}{\mu (B_{\epsilon})}$  as $\epsilon$  goes to 0.  ($B_\epsilon$ is the  ball  around $p$).


 But  I realized that the measure theory on  infinite  Banach  spaces is a technical matter:

https://mathoverflow.net/questions/36403/what-is-known-about-the-gaussian-measure-of-the-unit-ball-in-a-hilbert-space


https://mathoverflow.net/questions/404005/what-is-the-right-definition-of-zero-measure-subsets-of-banach-spaces


https://mathoverflow.net/questions/234249/invariant-probability-on-a-unit-ball-of-a-banach-space


https://math.stackexchange.com/questions/75932/measure-on-hilbert-space