### Abstract

Let n be a sufficiently large natural number and let B be an origin-symmetric convex body in R^{n} in the ℓ-position, and such that the space (R^{n},‖⋅‖_{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 E_{γn}‖⋅‖_{B}≤n^{c}E_{γn}‖grad_{B}(⋅)‖_{2} for a small universal constant c>0, where grad_{B}(⋅) is the gradient of ‖⋅‖_{B} and γ_{n} is the standard Gaussian measure in R^{n}. Then for any p∈[1,clogn] the p-th power of the norm ‖⋅‖_{B} ^{p} is [Formula presented]-superconcentrated in the Gauss space.

Original language | English (US) |
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Pages (from-to) | 121-151 |

Number of pages | 31 |

Journal | Journal of Functional Analysis |

Volume | 274 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2018 |

### All Science Journal Classification (ASJC) codes

- Analysis

### Keywords

- Almost Euclidean sections
- Dvoretzky's theorem
- Superconcentration
- ℓ-Position

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## Cite this

*Journal of Functional Analysis*,

*274*(1), 121-151. https://doi.org/10.1016/j.jfa.2017.08.021