Optimal Average Distortion for Non-Expansive Embeddings of Euclidean Metrics into the Line: With applications to cut-matching games and vertex expansion

Jack Spalding-Jamieson

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arXiv · 2610.41945

Optimal Average Distortion for Non-Expansive Embeddings of Euclidean Metrics into the Line: With applications to cut-matching games and vertex expansion

Jack Spalding-Jamieson

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Abstract

We show that any $n$-point Euclidean metric admits a non-expansive embedding into the line with average distortion at most $O\left(\sqrt{\frac{\log n}{\log\log n}}\right)$, matching a known lower bound. We also present two applications of this result. First, we give a cut-player strategy with an optimal spectral guarantee for the undirected cut-matching game. Second, we give an improved spectral approximation for vertex expansion in high-degree graphs.

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Jack Spalding-Jamieson. 2026-10-30. Optimal Average Distortion for Non-Expansive Embeddings of Euclidean Metrics into the Line: With applications to cut-matching games and vertex expansion. https://arxiv.org/abs/2610.41945

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