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Upper and lower bounds are derived for the Gaussian mean width of theintersection of a convex hull of $M$ points with an Euclidean ball of a givenradius. The upper bound holds for any collection of extreme point bounded inEuclidean norm. The upper bound and the lower bound match up to amultiplicative constant whenever the extreme points satisfy a one sidedRestricted Isometry Property.  This bound is then applied to study the Lasso estimator in fixed-designregression, the Empirical Risk Minimizer in the anisotropic persistenceproblem, and the convex aggregation problem in density estimation.

Localized Gaussian width of $M$-convex hulls with applications to Lasso and convex aggregation