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).
