Oracle inequalities and variable selection properties for the Lasso in linear models have been established under a variety of different assumptions on the design matrix. We show in this paper how the different conditions and concepts relate to each other. The restricted eigenvalue condition  or the slightly weaker compatibility condition  are sufficient for oracle results. We argue that both these conditions allow for a fairly general class of design matrices. Hence, optimality of the Lasso for prediction and estimation holds for more general situations than what it appears from coherence [5, 4] or restricted isometry  assumptions.
"On the conditions used to prove oracle results for the Lasso." Electron. J. Statist. 3 1360 - 1392, 2009. https://doi.org/10.1214/09-EJS506