In an article I read, I have the following inequality:
$\|A-B\|\_1 \geq \max \{ \|A 1\_m- B 1\_m \|\_1, \|A^T 1\_n - B^T1\_n\|\_1 \}$


Where $A, B \in \mathbb{R}\_+^{m\times n}$.
The $\|\cdot\|\_1$ refers either to the $l\_1$ vector norm, either to the matrix operator norm.
They use Jensen inequality for this result.


This result seems wrong in general, we can take $A = I\_2$ and $B = 0$ to show it.
I was wondering under what hypotheses this inequality is true.
Is it the case for bistochastic matrices ?



Norm inequality
The inequality is true if $\|A - B\|\_1$ is interpreted as $\sum |A\_{ij} - B\_{ij}|$.


Just observe that
$$ \|A 1\_M - B 1\_M\|\_1 = \sum\_i | \sum\_j A\_{ij} - B\_{ij}| \leq \sum\_i \sum\_j |A\_{ij} - B\_{ij}| $$
by the triangle inequality.


