Some applications of ball's extension theorem

We present two applications of Ball's extension theorem. First we observe that Ball's extension theorem, together with the recent solution of Ball's Markov type 2 problem due to Naor, Peres, Schramm and Sheffield, imply a generalization, and an alternative proof of, the Johnson-Lindenstrauss extension theorem. Second, we prove that the distortion required to embed the integer lattice {0,1,..., m} ⁿ, equipped with the ℓ _p ⁿ metric, in any 2-uniformly convex Banach space is of order min {n ^1/2-1/p, m ^1-2/p}.