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We study the asymptotic behaviour, as n → ∞, of the Lebesgue measure of the set $x\in K:|P_E\left(x\right)|\le t$ for a random k-dimensional subspace E ⊂ ℝⁿ and an isotropic convex body K ⊂ ℝⁿ. For k growing slowly to infinity, we prove it to be close to the suitably normalised Gaussian measure in ${ℝ}^k$ of a t-dilate of the Euclidean unit ball. Some of the results hold for a wider class of probabilities on ℝⁿ.

We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T\left[{({ℳ}_k,{\theta }_k)}_{k=1}^l\right]$ with index $i\left({ℳ}_k\right)$ finite are either $c_0$ or ${\ell }_p$ saturated for some $p$ and we characterize when any two spaces of such a form are totally incomparable in terms of the index $i\left({ℳ}_k\right)$ and the parameter ${\theta }_k$. Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T\left[{({𝒜}_k,{\theta }_k)}_{k=1}^{\infty }\right]$ in terms of the asymptotic behaviour of the sequence $∥{\sum }_{i=1}^n{e}_i∥$ where $\left(e_i\right)$ is the canonical basis....