TY - JOUR

T1 - Superconcentration, and randomized Dvoretzky's theorem for spaces with 1-unconditional bases

AU - Tikhomirov, Konstantin

N1 - Funding Information:
The research is partially supported by the Simons Foundation . I would like to thank Ramon van Handel for an interesting discussion. I am also grateful to Nicole Tomczak-Jaegermann, Alexander Litvak and Assaf Naor for valuable remarks.
Publisher Copyright:
© 2017 Elsevier Inc.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Let n be a sufficiently large natural number and let B be an origin-symmetric convex body in Rn in the ℓ-position, and such that the space (Rn,‖⋅‖B) admits a 1-unconditional basis. Then for any ε∈(0,1/2], and for random cεlogn/log[Formula presented]-dimensional subspace E distributed according to the rotation-invariant (Haar) measure, the section B∩E is (1+ε)-Euclidean with probability close to one. This shows that the “worst-case” dependence on ε in the randomized Dvoretzky theorem in the ℓ-position is significantly better than in John's position. It is a previously unexplored feature, which has strong connections with the concept of superconcentration introduced by S. Chatterjee. In fact, our main result follows from the next theorem: Let B be as before and assume additionally that B has a smooth boundary and Eγn‖⋅‖B≤ncEγn‖gradB(⋅)‖2 for a small universal constant c>0, where gradB(⋅) is the gradient of ‖⋅‖B and γn is the standard Gaussian measure in Rn. Then for any p∈[1,clogn] the p-th power of the norm ‖⋅‖B p is [Formula presented]-superconcentrated in the Gauss space.

AB - Let n be a sufficiently large natural number and let B be an origin-symmetric convex body in Rn in the ℓ-position, and such that the space (Rn,‖⋅‖B) admits a 1-unconditional basis. Then for any ε∈(0,1/2], and for random cεlogn/log[Formula presented]-dimensional subspace E distributed according to the rotation-invariant (Haar) measure, the section B∩E is (1+ε)-Euclidean with probability close to one. This shows that the “worst-case” dependence on ε in the randomized Dvoretzky theorem in the ℓ-position is significantly better than in John's position. It is a previously unexplored feature, which has strong connections with the concept of superconcentration introduced by S. Chatterjee. In fact, our main result follows from the next theorem: Let B be as before and assume additionally that B has a smooth boundary and Eγn‖⋅‖B≤ncEγn‖gradB(⋅)‖2 for a small universal constant c>0, where gradB(⋅) is the gradient of ‖⋅‖B and γn is the standard Gaussian measure in Rn. Then for any p∈[1,clogn] the p-th power of the norm ‖⋅‖B p is [Formula presented]-superconcentrated in the Gauss space.

KW - Almost Euclidean sections

KW - Dvoretzky's theorem

KW - Superconcentration

KW - ℓ-Position

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U2 - 10.1016/j.jfa.2017.08.021

DO - 10.1016/j.jfa.2017.08.021

M3 - Article

AN - SCOPUS:85029704732

VL - 274

SP - 121

EP - 151

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -