Constant-Distortion Embeddings of Hausdorff Metrics into Constant-Dimensional l_p Spaces

We show that the Hausdorff metric over constant-size pointsets in constant-dimensional Euclidean space admits an embedding into constant-dimensional `∞ space with constant distortion. More specifically for any s, d ≥ 1, we obtain an embedding of the Hausdorff metric over pointsets of size s in d-dimensional Euclidean space, into `s ∞ with distortion sO(s+d). We remark that any metric space M admits an isometric embedding into `∞ with dimension proportional to the size of M . In contrast, we obtain an embedding of a space of infinite size into constant-dimensional `∞. We further improve the distortion and dimension trade-offs by considering probabilistic embeddings of the snowflake version of the Hausdorff metric. For the case of pointsets of size s in the real line of bounded resolution, we obtain a probabilistic embedding into ` log s) 1 with distortion O(s). 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems