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

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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 normed space (Rⁿ,·B) admits a 1-unconditional basis. Then for any ε∈(0,1/2], and for random cε n/1/ε-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,c n] the p-th power of the norm ·Bᵖ is C/ n--superconcentrated in the Gauss space.