I want to find a condition for having all nonnegative $x_i$ in
$\min \bf{x}^T\bf{x}$
$s.t. \bf{Ax}=\bf{b}$
where $\bf{x}\in \mathbb{R}^{n\times 1}$, $\bf{A}\in \mathbb{R}^{m\times n}$, $\bf{b}\in \mathbb{R}^{m\times 1}$ and $m<n$.
From a geometric perspective, I can infer that if there exists a closed region formed by the hyperplane $\bf{Ax}=\bf{b}$ and the positive axes, the solutions will be all positive. But it is so abstract that I can't prove it using formulas and theorem.