 Lq null space property

For $0<q\le 1$, 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.