Fixed-sign null space condition
= Fixed-sign null space condition
For a real vector $x_0$ with support $S$, uniqueness in <basis pursuit> is equivalent to $|\langle\operatorname{sgn}(x_{0,S}),h_S\rangle|<\|h_{S^c}\|_1$ for every nonzero $h\in\ker A$. The supporting-line inequality for the <absolute value> proves sufficiency. Taking small positive and negative multiples of $h$ before any active sign changes proves necessity.