LOW M^{∗}-ESTIMATES ON COORDINATE SUBSPACES

APOSTOLOS A. GIANNOPOULOS VITALI D. MILMAN

Abstract. Let K be a symmetric convex body in R^{n}. It is well-known that for every θ ∈ (0, 1) there exists a subspace F of R^{n} with dimF = [(1 − θ)n] such that

(∗)  P_{F} (K) ⊇ \frac{c\sqrt{θ}}{M_{K}} D_{n} ∩ F,

where P_{F} denotes the orthogonal projection onto F. Consider a fixed coordinate system in R^{n}. We study the question whether an analogue of (∗) can be obtained when one is restricted to choose F among the coordinate subspaces R^{σ}, 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 of K. 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.