Let $ \boldsymbol{J} \in \mathbb{R}^{m \times n} $, with $ n>m $ be an invertible full row-rank matrix. Further: \begin{equation}\label{key} f= |\boldsymbol{J}^{-1} \boldsymbol{1}_{\pm}|_1 \end{equation} where $\boldsymbol{1}_{\pm} \in \mathbb{R}^{m \times 1}$ is a vector where each element is either 1 or minus 1, $ \boldsymbol{J}^{-1} $ is any inverse, and $|(\cdot)|_1$ is the absolute value norm.
Now, is it possible to find a factor $ c_1 $, (as function of properties of $ \boldsymbol{J} $) such that $f<c_1 $ for any $\boldsymbol{1}_{\pm}$. ?
Feel free to assert any new conditions on $\boldsymbol{J}$ to make this work.
Edit : If it makes it easier to solve, I am happy to receive answers where $\boldsymbol{1}_{\pm}|_1$ is simplified to $\boldsymbol{1}$ (i.e. a one-only vector).