Low M*-estimates on coordinate subspaces

A. A. Giannopoulos, V. D. Milman

Research output: Contribution to journalArticlepeer-review

Abstract

LetKbe a symmetric convex body inRn. It is well-known that for everyθ∈(0,1) there exists a subspaceFofRnwith dimF=[(1-θ)n] such thatPF(K)⊇cθMKDn∩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.

Original languageEnglish
Pages (from-to)457-484
Number of pages28
JournalJournal of Functional Analysis
Volume147
Issue number2
DOIs
StatePublished - Jul 1997

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