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The L¹ Poincaré constant of the Hamming cube

A mix of human-written and AI-generated textHuman understanding: all partsmath.PR — Probabilitymath.FA — Functional Analysismath.CO — Combinatorics

Contributed by Tejasvi Singh Tomar ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

Let CC be the smallest constant such that E∣f−Ef∣≤C E∣∇f∣\mathbb{E}|f-\mathbb{E} f|\le C\,\mathbb{E}|\nabla f| for every real-valued function ff on every Hamming cube {−1,1}n\{-1,1\}^n, where ∣∇f∣|\nabla f| is the Euclidean length of (D1f,…,Dnf)(D_1f,\dots,D_nf) and Djf(x)=(f(x)−f(x(j)))/2D_jf(x)=(f(x)-f(x^{(j)}))/2. Ben Efraim and Lust-Piquard proved C≤π/2C\le\pi/2, the Gaussian case gives C≥π/2C\ge\sqrt{\pi/2}, and Ivanisvili, Li, van Handel and Volberg asked whether C=π/2C=\sqrt{\pi/2}. We prove that it is, with an explicit constant cn<π/2c_n<\sqrt{\pi/2} in dimension nn. The proof turns on an integral of first absolute moments of sums of independent biased random signs: we compute it exactly when the coefficients are nonnegative, and show, with a sampled Fejér kernel, that it can only grow when the coefficients are replaced by their absolute values. The equality C=π/2C=\sqrt{\pi/2} is also proved in Lean 4 with Mathlib.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23715900

References

  1. L. Ben Efraim and F. Lust-Piquard, Poincaré type inequalities on the discrete cube and in the CAR algebra , Probab. Theory Related Fields 141 (2008), 569--602.
  2. P. Durcik, P. Ivanisvili and J. Roos, Sharp isoperimetric inequalities on the Hamming cube near the critical exponent , Adv. Math. 503 (2026), 111264; arXiv:2407.12674.arXiv
  3. P. Durcik, P. Ivanisvili, J. Roos and X. Xie, Sharp isoperimetric inequalities on the Hamming cube II: The critical exponent , arXiv:2602.20462, 2026.arXiv
  4. G. Han and P. Li, High-order Talagrand and Eldan--Gross inequalities via Besov-type variance functionals , arXiv:2606.14876, 2026.arXiv
  5. P. Ivanisvili, D. Li, R. van Handel and A. Volberg, Improving constant in end-point Poincaré inequality on Hamming cube , arXiv:1811.05584v5, 2019.arXiv
  6. P. Ivanisvili and Y. Stone, Sharpening the gap between L1L^1 and L2L^2 norms , arXiv:2407.04835, 2024.arXiv
  7. P. Ivanisvili, R. van Handel and A. Volberg, Rademacher type and Enflo type coincide , Ann. of Math. (2) 192 (2020), 665--678.
  8. P. Ivanisvili, X. Xie and H. Zhang, A Beckmann boundary form of Talagrand's conjecture on the discrete cube , arXiv:2606.31961, 2026.arXiv
  9. The mathlib Community, The Lean mathematical library , in: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2020), ACM, 2020, 367--381.
  10. A. W. Marshall, I. Olkin and B. C. Arnold, Inequalities: Theory of Majorization and Its Applications , 2nd ed., Springer, New York, 2011.
Show all 14 referencesShow fewer references
  1. L. de Moura and S. Ullrich, The Lean 4 theorem prover and programming language , in: Automated Deduction -- CADE 28, Lecture Notes in Comput. Sci. 12699 , Springer, 2021, 625--635.
  2. G. Pisier, Probabilistic methods in the geometry of Banach spaces , in: Probability and Analysis (Varenna, 1985), Lecture Notes in Math. 1206 , Springer, Berlin, 1986, 167--241.
  3. Y. Stone, The Poincaré inequality on the Hamming cube for vector-valued functions , Ph.D. thesis, University of California, Irvine, 2025.
  4. T. S. Tomar, An upper bound for the L1L^1 Poincaré constant of the Hamming cube , preprint (first version of this paper), 2026, doi:10.5281/zenodo.23524864 (https://doi.org/10.5281/zenodo.23524864).DOI

Version history

  1. v1 · 21 pagesSubmitted by Tejasvi Singh TomarPublished on Zenodo: https://doi.org/10.5281/zenodo.23715900CurrentNov 12, 2026