{"title":"How well do finite metric spaces embed in Euclidean space?","authors":"Leo Torres","abstract":"Bourgain proves that every metric space with $n$ points embeds in Euclidean space with distortion $O(\\log n)$, and constructs, from random graphs, metric spaces for which every embedding has distortion at least a constant multiple of $\\log n/\\log\\log n$. The lower bound shows in particular that $O(\\sqrt{\\log n})$ is not achievable, and answers negatively a question of Johnson and Lindenstrauss (1984) about embeddings into normed spaces of dimension $\\log n$. The embedding sends each point to its distances from random sets of points of every size up to $n$, and the analysis rests on a single estimate of how often such a set separates two given points. The theorem became a cornerstone of the theory of low-distortion embeddings of finite metric spaces. A decade later, Linial, London and Rabinovich (1995) made the construction algorithmic and proved that expander graphs require distortion of order $\\log n$, so that Bourgain's bound is best possible; with the independent work of Aumann and Rabani (1998), this made the embedding a basic tool of approximation algorithms, beginning with the sparsest cut problem.","keywords":["metric embeddings","distortion","Bourgain's theorem"],"year":2026,"slug":"bourgain-embedding","version":1,"url_hash":"00fe69e6c7c3","license":"cc-by-4.0","doi":"10.5281/zenodo.23931098","published_at":"2026-11-03T20:36:31.217849+00:00","subject":"Mathematics","storage_type":"archive","canonical_url":"https://scroll.press/2026/bourgain-embedding","version_url":"https://scroll.press/2026/bourgain-embedding/v1","paper_url":"https://scroll.press/2026/bourgain-embedding/paper","paper_version_url":"https://scroll.press/2026/bourgain-embedding/v1/paper","html_url":"https://scroll.press/2026/bourgain-embedding/paper","html_sha256":"00fe69e6c7c3b3b57cbd5d380c10bb9aa95e8343ede4eb06e528d566ba31d4aa","html_bytes":null,"cite_as":{"bibtex":"@misc{bourgain-embedding2026,\n  author = {Torres, Leo},\n  title = {How well do finite metric spaces embed in Euclidean space?},\n  year = {2026},\n  publisher = {Scroll Press},\n  url = {https://scroll.press/2026/bourgain-embedding},\n  version = {1},\n  doi = {10.5281/zenodo.23931098}\n}","csl_json":{"type":"article","id":"00fe69e6c7c3","title":"How well do finite metric spaces embed in Euclidean space?","author":[{"family":"Torres","given":"Leo"}],"publisher":"Scroll Press","URL":"https://scroll.press/2026/bourgain-embedding","version":"1","issued":{"date-parts":[[2026]]},"DOI":"10.5281/zenodo.23931098","license":"cc-by-4.0"}}}