TY - JOUR

T1 - Low M*-estimates on coordinate subspaces

AU - Giannopoulos, A. A.

AU - Milman, V. D.

N1 - Funding Information:
This work was initiated at the Institute for Advanced Study and was completed at the Mathematical Sciences Research Institute. Research at MSRI was supported in part by NSF Grant DMS-9022140.
Funding Information:
* Supported in part by BSF and NSF grants.

PY - 1997/7

Y1 - 1997/7

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0031188056&partnerID=8YFLogxK

U2 - 10.1006/jfan.1996.3054

DO - 10.1006/jfan.1996.3054

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AN - SCOPUS:0031188056

VL - 147

SP - 457

EP - 484

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -