LowM*-Estimates on Coordinate Subspaces

LetKbe a symmetric convex body inRn. It is well-known that for everyθ∈(0, 1) there exists a subspaceFofRnwith dim F=[(1−θ) n] such thatPF(K)⊇c θMK Dn∩F, ((*))where PFdenotes the orthogonal projection ontoF. Consider a fixed coordinate system inRn. We study the question whether an analogue of (*) can be obtained when one is restricted to chooseFamong the coordinate subspacesRσ,σ⊆{1, …, n}, with |σ|=[(1−θ) n]. We prove several “coordinate versions” of (*) in terms of the cotype-2 constant, of the volume ratio and other parameters ofK. The basic source of our estimates is an exact coordinate analogue of (*) in the ellipsoidal case. Applications to the computation of the number of lattice points inside a convex body are considered throughout the paper.