“…There is a central limit theorem due to Paouris, Pivovarov, and Zinn [18] for the volume of k-dimensional random projections of the n-dimensional cube when n → ∞, a result that had previously been obtained by Kabluchko, Litvak, and Zaporozhets [12] in the special case k = 1. Alonso-Gutiérrez, Prochno, and Thäle [1] proved a central limit theorem and Berry-Esseen bounds for the Euclidean norm of random orthogonal projections of points chosen uniformly at random from the unit ball of ℓ n p , as n → ∞, and Kabluchko, Prochno, and Thäle [13] obtained a multivariate central limit theorem for the q-norm of random vectors chosen uniformly at random in the unit p-ball of R n , which extended the corresponding 1-dimensional result obtained by Schmuckenschläger [22]. While the results in the previous paragraph describe central limit phenomena for several geometry related quantities, there is considerably less known about the large deviations behavior.…”