In previous work, numerical experiments showed that ℓ^p minimization with 0 < p < 1 recovers sparse signal from fewer linear measurements than does ℓ¹ minimization. It was also shown that a weaker restricted isometry property is sufficient to guarantee perfect recovery in the ℓ^p case. In this work, we generalize this result to an ℓ^p of the restricted isometry property, and then determine how many random, Gaussian measurements are sufficient for the condition to hold with high probability. The resulting sufficient condition is met by fewer measurements for smaller p.