A Non-Probabilistic Proof of the Assouad Embedding Theorem with Bounds on the Dimension

Abstract

On redémontre de manière constructive un résultat de Naor and Neiman (à paraître, Revista Matematica Iberoamercana), qui dit que pour tout espace métrique doublant (E, d), il existe N ≥ 0, qui ne dépend que de la constante de doublement, tel que pour tout exposant α ∈]1/2, 1[, il existe une application bilipschitzienne F de (E, d α )à valeurs dans R N .Abstract. We give a constructive proof of a theorem of Naor and Neiman, (to appear, Revista Matematica Iberoamercana), which asserts that if (E, d) is a doubling metric space, there is an integer N > 0, that depends only on the metric doubling constant, such that for each exponent α ∈ (1/2, 1), we can find a bilipschitz mapping F = (E, d α ) → R N .The purpose of this paper is to give a simpler and constructive proof of a theorem proved by Naor and Neiman [NN], which asserts that if (E, d) is a doubling metric space, there is an integer N > 0, that depends only on the metric doubling constant, such that for each exponent α ∈ (1/2, 1), we can find a bilipschitz mapping F = (E, d α ) → R N .Here R N is equipped with its Euclidean metric, the snowflake distance d α is simply defined by d α (x, y) = d(x, y) α for x, y ∈ E, and metrically doubling means that there is an integer C 0 ≥ 1 such that for every r > 0, every (closed) ball of radius 2r in E can be covered with no more than C 0 balls of radius r. We call C 0 a metric doubling constant for (E, d).

Metric Spaces Admitting Low-distortion Embeddings into All n-dimensional Banach Spaces

“…Recently, Naor and Neiman [NN12] (cf. also [DS13]) obtained estimates of N depending only on the doubling constant and independent of α, for α ∈ (1/2, 1). Note that the doubling condition is important for 2 -embeddability in Theorem 3.9.…”

“…One of the most important results on this problem is the Assouad theorem, which we mention below (Theorem 3.9). However, it should be mentioned that in the present context we are interested in the version of the problem for which the dimension is specified, see very interesting recent results related to this problem in [NN12,DS13], and related comments in [Hei03, Remark 3.16].…”

Metric Dimension Reduction: A Snapshot of the Ribe Program

“…Characterizing the quantitative dependence in terms of geometric properties of the target norm on R k has not been carried out yet (it isn't even clear what should the pertinent geometric properties be), though see [114] for an almost isometric version when one considers the ∞ norm on R k (with the dimension k tending to ∞ as the distortion approaches 1); see also [103] for a further partial step in this direction. In [191] it was shown that for θ ∈ [ 1 2 , 1) one could take k(K, θ) k(K) to be bounded by a constant that depends only on K; the proof of this fact in [191] relies on a probabilistic construction, but in [77] a clever and instructive deterministic proof of this phenomenon was found (though, yielding asymptotically worse estimates on α(K, θ), k(K) than those of [191]).…”

“…6 On its own, the established necessity of obtaining a genuinely nonlinear embedding method into low dimensions should not discourage attempts to answer Question 41, because some rigorous nonlinear dimension reduction methods have been devised in the literature; see e.g. [24,233,63,110,32,33,139,146,59,154,62,1,34,112,191,77,150,203,103,30,201,206,16]. However, all of these approaches seem far from addressing Question 41.…”

Isometric embeddings of snowflakes into finite-dimensional Banach spaces

“…A metric space quasisymmetrically embeds into some Euclidean spaces if and only if it is doubling (see [Hei01,Theorem 12.1]). More specifically, Assouad proved the following result (see [Ass83], and also [NN12,DS13]): if (X, d) is a doubling metric space and α ∈ (0, 1) then the metric space (X, d α ) admits a bi-Lipschitz embedding into some Euclidean space. If d E is the Euclidean distance and α = log 2/ log 3 then the metric space ([0, 1], d α E ) is bi-Lipschitz equivalent to the von Koch snowflake curve, so (X, d α ) is said to be the α-snowflake of (X, d).…”