  Upper and lower bounds are derived for the Gaussian mean width of the
intersection of a convex hull of $M$ points with an Euclidean ball of a given
radius. The upper bound holds for any collection of extreme point bounded in
Euclidean norm. The upper bound and the lower bound match up to a
multiplicative constant whenever the extreme points satisfy a one sided
Restricted Isometry Property.
  This bound is then applied to study the Lasso estimator in fixed-design
regression, the Empirical Risk Minimizer in the anisotropic persistence
problem, and the convex aggregation problem in density estimation.
