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In this paper, we study high-dimensional random projections of ℓ n p -balls. More precisely, for any n ∈ N let En be a random subspace of dimension kn ∈ {1, . . . , n} and Xn be a random point in the unit ball of ℓ n p . Our work provides a description of the Gaussian fluctuations of the Euclidean norm PE n Xn 2 of random orthogonal projections of Xn onto En. In particular, under the condition that kn → ∞ it is shown that these random variables satisfy a central limit theorem, as the space dimension n tends to infinity. Moreover, if kn → ∞ fast enough, we provide a Berry-Esseen bound on the rate of convergence in the central limit theorem. At the end we provide a discussion of the large deviations counterpart to our central limit theorem.2010 Mathematics Subject Classification. Primary: 52A22 Secondary: 46B07, 60F05.

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Abstract. Let K be the symmetric convex hull of m independent random vectors uniformly distributed on the unit sphere of R n . We prove that, for every δ > 0, the isotropy constant of K is bounded by a constant c(δ) with high probability, provided that m ≥ (1 + δ)n.

We prove various extensions of the Loomis-Whitney inequality and its dual, where the subspaces on which the projections (or sections) are considered are either spanned by vectors w i of a not necessarily orthonormal basis of R n , or their orthogonal complements. In order to prove such inequalities we estimate the constant in the Brascamp-Lieb inequality in terms of the vectors w i . Restricted and functional versions of the inequality will also be considered.1 k ,