Given $A\in\Bbb R^{m\times n}$, define $$\gamma_2(A)=\min_{XY=A}\max_{i,j}\|x_i\|_{\ell_2}\|y_j\|_{\ell_2}$$ where rows of $X$,columns of $Y$ given by $\{x_i\}_{i=1}^m,\{y_j\}_{j=1}^n$ respectively.

Where could I find a reference for fact that $$\gamma_2^2(A)\leq\| A\|_{\ell_1^n\rightarrow\ell_{\infty}^m}rank(A)?$$

Should this be $$\gamma_2^2(A)\leq\| A\|_{\ell_1^n\rightarrow\ell_{\infty}^m}^2rank(A)?$$

If $A\in\{\pm1\}^{m\times n}$, is there a method through which above reduces to $$\gamma_2^2(A)\leq rank(A)?$$

Above is lemma 4.2 in http://www2.mta.ac.il/~adish/Pubs/Papers/complexity_matrices.pdf

I do not understand this proof. Is there an easy technique to understand this proof?

What is terminology for $\gamma_2(A)$ in literature?

posted http://math.stackexchange.com/questions/1196950/a-question-in-banach-space