Under the usual definition, the <null space property> relative to $S$ is $\|h_S\|_1<\|h_{S^c}\|_1$ for every nonzero $h\in\ker A$. It characterizes recovery by <basis pursuit> of every vector supported in $S$. Indeed, if it holds, then for any such $x_0$ and any nonzero $h\in\ker A$,
$$
\|x_0+h\|_1\ge\|x_0\|_1-\|h_S\|_1+\|h_{S^c}\|_1>\|x_0\|_1.
$$
Conversely, failure for some $h$ makes $x_0=-h_S$ and $z=h_{S^c}$ feasible with the same measurements and $\|z\|_1\le\|x_0\|_1$. Thus
$$
\boxed{\text{uniform recovery on }S\ \Longleftrightarrow\ \text{null space property on }S.}
$$
For the single fixed vector in the printing, necessity is false with that definition. For example,
$$
A=\begin{pmatrix}1&-1&0\\1&0&-1\end{pmatrix},\quad x_0=(1,-1,0),\quad S=\{1,2\}
$$
has $\ker A=\operatorname{span}\{(1,1,1)\}$. Along its feasible line, the objective is $|1+t|+|-1+t|+|t|$, uniquely minimized at $t=0$, although the <null space property> would require $2<1$.

The correct <fixed-sign null space condition> is
$$
\boxed{|\langle\operatorname{sgn}(x_{0,S}),h_S\rangle|<\|h_{S^c}\|_1\quad(0\ne h\in\ker A).}
$$
Sufficiency follows from the supporting-line inequality for the <absolute value>. Necessity follows by considering $x_0\pm th$ for small positive $t$, when all signs on $S$ remain unchanged: a nonpositive directional increase gives either a decrease or another minimizer. If “relative to $S$” was intended to include these signs, this is the precise condition needed.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 340 / 4 / a / Solution

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.

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 Incoming links: Fixed-sign null space condition