Remarks on the geometry of coordinate projections in ℝⁿ

We study geometric properties of coordinate projections. Among other results, we show that if a body K ⊂ ℝⁿ has an "almost extremal" volume ratio, then it has a projection of proportional dimension which is close to the cube. We also establish a sharp estimate on the shattering dimension of the convex hull of a class of functions in terms of the shattering dimension of the class itself.