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

Let n be a sufficiently large natural number and let B be an origin-symmetric convex body in Rnin 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 cεlog⁡n/log⁡1ε-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,clog⁡n] the p-th power of the norm ‖⋅‖Bp is Clog⁡n-superconcentrated in the Gauss space.