Extensions of lipschitz maps into Banach spaces

Polynomial-Sized Topological Approximations Using the Permutahedron

“…To adapt the aforementioned schemes to play nice with high dimensional point clouds, it makes sense to use dimension reduction results to eliminate the dependence on λ. Indeed, it has been shown, in analogy to the famous Johnson-Lindenstrauss Lemma [18], that an orthogonal projection of a point set of R d to a O(log n/ε 2 )-dimensional subspace yields a (1 + ε) approximation of theČech filtration [20,29]. Combining these two approximation schemes, however, yields an approximation of size O(n k+1 ) (ignoring ε-factors) and does not improve upon the exact case.…”

“…Proof The famous lemma of Johnson and Lindenstrauss [18] asserts the existence of a map f as in Lemma 6.2 for m = λ log n/ε 2 with some absolute constant λ and ξ 1 = (1 − ε), ξ 2 = (1 + ε). Choosing ε = 1/2, we obtain that m = O(log n) and ξ 2 /ξ 1 = 3.…”

On Lipschitz extension from finite subsets

“…In the end, the entire argument as presented in Section 3 uses geometric and combinatorial considerations that are self-contained and do not make any reference to the clustering objective OPT (G, w, d T ). The examples of metric spaces that we use to prove Theorem 1 are modifications of the examples that were considered in [CKR05], which are themselves in the spirit of an example that was used in [JLS86] (and, we use yet another variant of the example of [JLS86] to prove Theorem 4).…”

“…In [JL84] it was shown that ae(ε) 1/ε, and a different and very simple proof of this fact was given in [JLS86]. In [MN13a] it was proved that ae(ε) 1/ε, via a quantitative refinement of a beautiful argument of Kalton [Kal12].…”

“…We shall also use the notation A ≍ B when A B and A B. The above upper bound of [JLS86] on ae(n) was more recently improved [LN04,LN05] to ae(n) (log n)/ log log n, which is the best-known upper bound on ae(n) to date. Determining the asymptotic behavior of ae(n) as n → ∞ remains a major open problem.…”