Bourgain's Theorem for a Union of Euclidean Metric Spaces
Abstract
We study the distortion of embedding a union of k Euclidean sets into Euclidean space. We show that a union of k Euclidean sets can be embedded with distortion at most O(k1.128 log k), independent of the number of points. This is an exponential improvement over the previous best known upper bound of 2O(k) obtained by iterating the result of Makarychev and Makarychev (2016) for the special case k = 2. Surprisingly, with the additional assumption that each Euclidean set remains Euclidean when we adjoin any other two points, we can obtain an asymptotically tight distortion of O(log k), thus, providing a generalization of Bourgain's Theorem (1985). Along the way, we also show that the distortion of the union can be upper bounded by O(log k) times the maximum of the Lipschitz extension constants of individual Euclidean sets.
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BibTeX
@misc{chakarov2026bourgains,
title = {{Bourgain's Theorem for a Union of Euclidean Metric Spaces}},
author = {Dimitar Chakarov and Sepideh Mahabadi and Yury Makarychev and Ali Vakilian},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08546},
eprint = {2611.08546},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08546},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bourgain's Theorem for a Union of Euclidean Metric Spaces
AU - Dimitar Chakarov
AU - Sepideh Mahabadi
AU - Yury Makarychev
AU - Ali Vakilian
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08546
UR - https://arxiv.org/abs/2611.08546
AB - We study the distortion of embedding a union of k Euclidean sets into Euclidean space. We show that a union of k Euclidean sets can be embedded with distortion at most O(k^1.128 log k), independent of the number of points. This is an exponential improvement over the previous best known upper bound of 2^O(k) obtained by iterating the result of Makarychev and Makarychev (2016) for the special case k = 2. Surprisingly, with the additional assumption that each Euclidean set remains Euclidean when we adjoin any other two points, we can obtain an asymptotically tight distortion of O(log k), thus, providing a generalization of Bourgain's Theorem (1985). Along the way, we also show that the distortion of the union can be upper bounded by O(log k) times the maximum of the Lipschitz extension constants of individual Euclidean sets.
ER -