Does there exist a sequence with $ \int_{[0,1]^2}| f_i - f_j |\, {\rm d}x = {\rm const.} > 0 $ for $ \forall i \neq j $? Let $A = [0,1]^2$. Is there a such a sequence of functions $ f_i\colon A \rightarrow \mathbb{R} $ with $ \int_{A}| f_i | \ {\rm d}x  \leqslant 1$ and $ \int_{A} | f_i - f_j | \ {\rm d}x =  {\rm const.} > 0 $ for $ \forall i \neq j $ ? $ x \in \mathbb{R}^2 $
I have found this thread here: Is this a totally bounded set in the space of continuous functions?
, which was very interesting, but not yet enough to give an example. At the moment I am still trying to figure out, whether such a sequence even exists. 
So I would be very happy for any constructive hint, comment, answer or recommendation for further reading. Thanks in advance. 

If $\mu$ is an atomless measure then $L_1(\mu)$ contains an isometric copy of $\ell_2$ (actually of any $\ell_p$ for $p\in [1,2]$); for example take a sequence of independent Gaussian random variables, their span is isometric to $\ell_2$. Certainly, it is easy to find such a set in $\ell_2$ (any orthonormal basis would do). In conclusion, you can get $c=2^{1/p}$ for any $p\in [1,2]$.
