 Solution

Let $\sigma =\mathrm{sgn}(x_S)$, using the real sign function on the support of a vector $x$. The fixed-sign null space condition in the PDF uses $S^c$ on its right-hand side. This complement is essential. For any real coordinate $x_j\ne 0$, the supporting-line inequality for the absolute value isEvery distinct feasible vector is $x+v$ with $0\ne v\in \ker A$. Summing the coordinate inequalities on $S$ and adding the L1 norm on $S^c$ givesThe strict final inequality is precisely the fixed-sign null space condition. Hence $x$ is the unique basis pursuit minimizer. Unlike the null space property of Question 1, this condition concerns the particular sign function values of $x$, rather than every vector supported in $S$.