  This article investigates a new parameter for the high-dimensional regression
with noise: the distortion. This latter has attracted a lot of attention
recently with the appearance of new deterministic constructions of
'almost'-Euclidean sections of the L1-ball. It measures how far is the
intersection between the kernel of the design matrix and the unit L1-ball from
an L2-ball. We show that the distortion holds enough information to derive
oracle inequalities (i.e. a comparison to an ideal situation where one knows
the s largest coefficients of the target) for the lasso and the Dantzig
selector.
