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PreprintCC BY 4.0arXiv

Optimal MM* bounds for convex bodies

  • Pierre Bizeul
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Abstract

Let K⊂Rn be a convex body in isotropic position. We prove the optimal mean-width estimate M*(K)≤ C√(nlog n). The main new ingredient is a geometric inequality relating the Gaussian mean of the support function to its mean under the uniform measure on K, obtained through a heat-flow argument. Combined with the Gaussian-log-concave comparison of Eldan and Lehec and the newly available dimension-free bound on the third-moment parameter κn, this yields the result. The boundedness of κn also makes the mean-norm estimate of Bizeul and Klartag sharp. Combining both estimates yields M(K)M*(K)≤ Clog n, extending Pisier's MM* estimate to non-symmetric convex bodies and to the isotropic position.

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Cite

BibTeXRIS
BibTeX
@misc{bizeul2026optimal,
  title = {{Optimal MM* bounds for convex bodies}},
  author = {Pierre Bizeul},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.29458},
  eprint = {2607.29458},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.29458},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Optimal MM* bounds for convex bodies
AU  - Pierre Bizeul
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2607.29458
UR  - https://arxiv.org/abs/2607.29458
AB  - Let K⊂R^n be a convex body in isotropic position. We prove the optimal mean-width estimate M^*(K)≤ C√(nlog n). The main new ingredient is a geometric inequality relating the Gaussian mean of the support function to its mean under the uniform measure on K, obtained through a heat-flow argument. Combined with the Gaussian-log-concave comparison of Eldan and Lehec and the newly available dimension-free bound on the third-moment parameter κ_n, this yields the result. The boundedness of κ_n also makes the mean-norm estimate of Bizeul and Klartag sharp. Combining both estimates yields M(K)M^*(K)≤ Clog n, extending Pisier's MM^* estimate to non-symmetric convex bodies and to the isotropic position.
ER  -