TY - JOUR

T1 - Duality of metric entropy in Euclidean space

AU - Artstein, Shiri

AU - Milman, Vitali D.

AU - Szarek, Stanislaw J.

N1 - Funding Information:
This research was partially supported by grants from the US-Israel BSF (all authors) and the NSF [USA] (the third named author).

PY - 2003/12/1

Y1 - 2003/12/1

N2 - Let K be a convex body in a Euclidean space, K○ its polar body and D the Euclidean unit ball. We prove that the covering numbers N (K, t D) and N (D, t K○) are comparable in the appropriate sense, uniformly over symmetric convex bodies K, over t > 0 and over the dimension of the space. In particular this verifies the duality conjecture for entropy numbers of linear operators, posed by Pietsch in 1972, in the central case when either the domain or the range of the operator is a Hilbert space.

AB - Let K be a convex body in a Euclidean space, K○ its polar body and D the Euclidean unit ball. We prove that the covering numbers N (K, t D) and N (D, t K○) are comparable in the appropriate sense, uniformly over symmetric convex bodies K, over t > 0 and over the dimension of the space. In particular this verifies the duality conjecture for entropy numbers of linear operators, posed by Pietsch in 1972, in the central case when either the domain or the range of the operator is a Hilbert space.

UR - http://www.scopus.com/inward/record.url?scp=0347534116&partnerID=8YFLogxK

U2 - 10.1016/j.crma.2003.09.033

DO - 10.1016/j.crma.2003.09.033

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AN - SCOPUS:0347534116

VL - 337

SP - 711

EP - 714

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

SN - 1631-073X

IS - 11

ER -