We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into c₀ and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the best constant for Lipschitz embeddings of the classical ${\ell }_p$-spaces into c₀ and give other applications. We prove that if a Banach space embeds almost isometrically into c₀, then it embeds linearly almost isometrically into c₀. We also study Lipschitz embeddings into...

Bourgain’s discretization theorem asserts that there exists a universal constant $C\in (0,\infty )$ with the following property. Let $X,Y$ be Banach spaces with $dimX=n$. Fix $D\in (1,\infty )$ and set $\delta =e^{-n^{Cn}}$. Assume that $𝒩$ is a $\delta$-net in the unit ball of $X$ and that $𝒩$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $D$. Then the entire space $X$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $CD$. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain’s theorem.We also obtain an improvement...