Two observations regarding embedding subsets of Euclidean spaces in normed spaces

This paper contains two results concerning linear embeddings of subsets of Euclidean space in low-dimensional normed spaces. The first is an improvement of the known dependence on ɛ in Dvoretzky's theorem from order of ɛ2 to order of ɛ (except for log factors). The second is a joint generalization of (Milman's version of) Dvoretzky's theorem and (a recent generalization by Klartag and Mendelson of) the Johnson–Lindenstrauss Lemma.