Abstract. Let ε > 0 and 1 ≤ k ≤ n and let {W l } p l=1 be affine subspaces of R n , each of dimension at most k. Let m = O(ε −2 (k + log p)) if ε < 1, and m = O(k + log p/log(1 + ε)) if ε ≥ 1. We prove that there is a linear map H : R n → R m such that for all 1 ≤ l ≤ p and x, y ∈ W l we have x − y 2 ≤ H(x) − H(y) 2 ≤ (1 + ε) x − y 2, i.e. the distance distortion is at most 1 + ε. The estimate on m is tight in terms of k and p whenever ε < 1, and is tight on ε, k, p whenever ε ≥ 1. We extend these results to embeddings into general normed spaces Y .

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