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 $\{0,1,\ldots,m\}^n$, equipped with the $\ell_p^n$ metric, in any $2$-uniformly convex Banach space is of order $\min \left\{n^{\frac12-\frac{1}{p}},m^{1-\frac{2}{p}}\right\}$.