Let $M$ be a real $n\times n$ matrix and let $\|M\|_1 = \sum_{ij} |M_{ij}|$ ("the entry-wise 1-norm").
My question is: How well can we bound $\|M\|_1 $ in terms of $|\det M|$ (from below)?
Here is an obvious bound: Let $m = \operatorname{max}_{ij} |M_{ij}|$.
Then it is obvious that $|\det M| \leq \sum_{\sigma \in S_n} m^n = n! m^n \quad or \quad \|M\|_1 \geq m \geq (|\det M|/n!)^{\frac1n}.$ Can this bound be improved?
(In fact, the above result is strong enough for my purposes, but this question came out of curiosity. I have played around for a while but I couldn't improve the bound, nor achieve it.)