The Restricted Isometry Property (RIP) is a fundamental property of a matrix which enables sparse recovery. Informally, an $m \times n$ matrix satisfies RIP of order $k$ for the $\ell_p$ norm, if $\|Ax\|_p \approx \|x\|_p$ for every vector $x$ with at most $k$ non-zero coordinates. For every $1 \leq p < \infty$ we obtain almost tight bounds on the minimum number of rows $m$ necessary for the RIP property to hold. Prior to this work, only the cases $p = 1$, $1 + 1 / \log k$, and $2$ were studied. Interestingly, our results show that the case $p = 2$ is a "singularity" point: the optimal number of rows $m$ is $\widetildeΘ(k^{p})$ for all $p\in [1,\infty)\setminus \{2\}$, as opposed to $\widetildeΘ(k)$ for $k=2$. We also obtain almost tight bounds for the column sparsity of RIP matrices and discuss implications of our results for the Stable Sparse Recovery problem.

An extended abstract of this paper is to appear at the 31st International Symposium on Computational Geometry (SoCG 2015)