= Lq null space property
{c}
{title2=$\sum_{i\in S}|v_i|^q<\sum_{i\notin S}|v_i|^q$}
For $0<q\le1$, this strict inequality for every nonzero $v\in\ker A$ and every $|S|\le s$ is equivalent to unique constrained $\ell^q$ recovery of every <sparse vector> of order $s$. For sufficiency, use $|x_i+v_i|^q\ge|x_i|^q-|v_i|^q$ on the <support of a vector>. For necessity compare $x=-v_S$ with $v_{S^c}$, which have the same measurements. The statement concerns uniform recovery over the whole sparse class, rather than a single signed <vector>.
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