I will begin with the definitions of Restricted Isometry Property (RIP) and Robust Null Space Property (RNSP), which are the two widely used sufficient conditions for robust sparse recovery in the field of compressed sensing.
Definition 1: A matrix $A \in \mathbb{R}^{m\times n}$ is said to satisfy the restricted isometry property (RIP) of order $k$ with constant $\delta_k \in (0,1)$ if $$ (1 - \delta_k) ||u||_2^2 \leq || Au||_2^2\leq (1 + \delta_k) ||u||_2^2 ,\forall u\ \mbox{such that}\ ||u||_0\leq k. $$
Definition 2: A matrix $A \in \mathbb{R}^{m\times n}$ is said to satisfy the robust null space property (RNSP) of order $k$ with constants $\rho \in (0,1)$ and $\tau \in \mathbb{R}_+$, if for all $h \in \mathbb{R}^n$ and all $S \subset \{1,\ 2,...,\ n\}$ with $|S| \leq k$, it is true that $$ ||h_S||_1 \leq \rho ||h_{S^c} ||_1 + ||Ah||_2 . $$
Now, I present the theorem which connects the RIP to the RNSP.
Theorem 1 : Suppose that, for some number $t > 1$, the matrix $A$ satisfies the RIP of order $tk$ with constant $\delta_{tk} =: \delta < \sqrt{ (t-1)/t }$. Define $$ \nu := \sqrt{t(t-1)} - (t-1) \in (0,0.5) , $$ $$ a := [ \nu ( 1 - \nu) - \delta ( 0.5 - \nu + \nu^2) ]^{1/2}, $$ $$ b := \nu ( 1 - \nu ) \sqrt{1+\delta} , c := \left[ \frac { \delta \nu^2 }{ 2(t-1) } \right]^{1/2} . $$ Then $A$ satisfies the RNSP with constants. $$ \rho = \frac{c}{a} < 1 ,\ \tau = \frac{b \sqrt{k} }{a^2}. $$ Based on the above theorem, the author says: $$\textbf{RIP}\implies \textbf{RNSP}$$ And hence, RNSP is a weaker sufficient condition than the RIP.
Now, I understand how RIP implies RNSP. But I don't understand what the significance of the statement that RNSP is a weaker sufficient condition. What does it signify that RNSP is a weaker sufficient condition than the RIP ? Does it mean RNSP is a less restrictive condition than the RIP?