A polynomial bound in Dvoretzky's theorem

Abstract

We present a simple proof of the ε-Dvoretzky conjecture, which asserts that the dependence on the approximation parameter ε in Dvoretzky's theorem is polynomial in 1/ε. In particular, if n ≥ (C/ε)^ℓ/2+1, then any n-dimensional convex body has, through any given interior point, an ℓ-dimensional section that is ε-close to a Euclidean ball. Here, C > 0 is a universal constant. We in fact obtain a sharper dependence on ε. The proof is probabilistic, but uses a different probabilistic model from those employed previously. We also prove a simultaneous version of our theorem for a finite family of convex bodies with the origin in their interior, yielding a common linear subspace on which all of the bodies have nearly-Euclidean sections. Finally, we compare the radius of the approximating Euclidean ball with familiar geometric parameters, such as the mean widths of the body and its dual.

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BibTeX RIS

BibTeX

@misc{klartag2026polynomial,
  title = {{A polynomial bound in Dvoretzky's theorem}},
  author = {Boaz Klartag and Shahar Moshe},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03204},
  eprint = {2610.03204},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03204},
}

RIS

TY  - UNPB
TI  - A polynomial bound in Dvoretzky's theorem
AU  - Boaz Klartag
AU  - Shahar Moshe
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03204
UR  - https://arxiv.org/abs/2610.03204
AB  - We present a simple proof of the ε-Dvoretzky conjecture, which asserts that the dependence on the approximation parameter ε in Dvoretzky's theorem is polynomial in 1/ε. In particular, if n ≥ (C/ε)^ℓ/2+1, then any n-dimensional convex body has, through any given interior point, an ℓ-dimensional section that is ε-close to a Euclidean ball. Here, C > 0 is a universal constant. We in fact obtain a sharper dependence on ε. The proof is probabilistic, but uses a different probabilistic model from those employed previously. We also prove a simultaneous version of our theorem for a finite family of convex bodies with the origin in their interior, yielding a common linear subspace on which all of the bodies have nearly-Euclidean sections. Finally, we compare the radius of the approximating Euclidean ball with familiar geometric parameters, such as the mean widths of the body and its dual.
ER  -

Abstract

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