Robust null space property ($\Vert h_S{\Vert }_2\le \rho \Vert h_{S^c}{\Vert }_1/\sqrt{s}+\tau \Vert Ah{\Vert }_2$)

The displayed property holds for every $h$ and every $|S|\le s$, with $0\le \rho <1$ and $\tau >0$. It converts the cone inequality from basis pursuit minimality and the tube inequality from noisy feasibility into a two-constant bound $\Vert \hat{x}-x_0{\Vert }_2\le C_1\eta +C_2{\sigma }_s(x_0)/\sqrt{s}$. Both constants are needed in general; a fixed coefficient one on the approximation term does not follow.