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Dimensionality of Hamming Metrics and Rademacher Type

Author Alexandros Eskenazis

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Alexandros Eskenazis
  • CNRS, Institut de Mathématiques de Jussieu, Sorbonne Université, France
  • Trinity College, University of Cambridge, UK

Acknowledgements

I wish to thank Florent Baudier, Paata Ivanisvili and Assaf Naor for their constructive feedback on this work.

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Alexandros Eskenazis. Dimensionality of Hamming Metrics and Rademacher Type. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.55

Abstract

Let X be a finite-dimensional normed space. We prove that if the Hamming cube {-1,1}ⁿ embeds into X with bi-Lipschitz distortion at most D ≥ 1, then dim(X) ≳ sup_{p ∈ [1,2]} n^p/(D^p 𝖳_p(X)^p), where 𝖳_p(X) is the Rademacher type p constant of X. This estimate yields a mutual refinement of distortion lower bounds which follow from works of Oleszkiewicz (1996) and Ivanisvili, van Handel and Volberg (2020). The proof relies on a combination of semigroup techniques on the biased hypercube with the Borsuk-Ulam theorem from algebraic topology.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Functional analysis
  • Theory of computation → Random projections and metric embeddings
Keywords
  • Hamming cube
  • Rademacher type
  • metric embeddings
  • Borsuk-Ulam theorem

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