LOW M-ASTERISK-ESTIMATES ON COORDINATE SUBSPACES

摘要

Let K be a symmetric convex body in R-n. It is well-known that for every theta is an element of (0, 1) there exists a subspace F of R-n with dim F = [(1 - theta) n] such that P-F(K) superset of or equal to c root theta/M-K D-n boolean AND 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-sigma, sigma subset of or equal to {1, ..., n}, with sigma = [(1 - theta) 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. (C) 1997 Academic Press. [References: 27]