Norm inequality

Question

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 ?

Answer 1

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.