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Let $n$ be a sufficiently large natural number and let $B$ be anorigin-symmetric convex body in $R^n$ in the $\ell$-position, and such that thenormed space $(R^n,\|\cdot\|_B)$ admits a $1$-unconditional basis. Then for any$\varepsilon\in(0,1/2]$, and for random $c\varepsilon\logn/\log\frac{1}{\varepsilon}$-dimensional subspace $E$ distributed according tothe rotation-invariant (Haar) measure, the section $B\cap E$ is$(1+\varepsilon)$-Euclidean with probability close to one. This shows that the"worst-case" dependence on $\varepsilon$ in the randomized Dvoretzky theorem inthe $\ell$-position is significantly better than in John's position. It is apreviously unexplored feature, which has strong connections with the concept ofsuperconcentration introduced by S. Chatterjee. In fact, our main resultfollows from the next theorem: Let $B$ be as before and assume additionallythat $B$ has a smooth boundary and ${\mathbb E}_{\gamma_n}\|\cdot\|_B\leqn^c\,{\mathbb E}_{\gamma_n}\big\|{\rm grad}_B(\cdot)\big\|_2$ for a smalluniversal constant $c>0$, where ${\rm grad}_B(\cdot)$ is the gradient of$\|\cdot\|_B$ and $\gamma_n$ is the standard Gaussian measure in $R^n$. Thenfor any $p\in[1,c\log n]$ the $p$-th power of the norm $\|\cdot\|_B^p$ is$\frac{C}{\log n}$--superconcentrated in the Gauss space.

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