Optimal mean width and metric entropy estimates for convex bodies
Abstract
We show that for any n ≥ 1 and any convex body K ⊂ Rn, there exists T ∈ SL(n) such that diam(TK) ≲ √n vr(K), and M∗(T(K−x) ∩ rvr(K) Bn2)≲ vr(K) √(log(e r2)), for every x ∈ K and every r ≥ 1. Above, M∗(·) denotes the spherical mean width and vr(·) denotes the volume radius. As a consequence, we establish for any convex body K ⊂ Rn that 1 ≤ infT ∈ SL(n) M∗(TK)/vr(K) ≲ √(log(e n)). The estimates above are sharp, up to universal constants, as they are attained for the crosspolytope and any regular n-simplex. Up to universal constants, our results show that all quermassintegrals, and the logarithm of the covering numbers for all scales, simultaneously for K and the polar body K∘, are maximized by the simplex and crosspolytope. Our proof makes use of Eldan's stochastic localization. To establish the results, we work with an extension of Bobkov's maximal Gaussian measure position to possibly non-symmetric convex bodies. Our results imply that this position is an optimal "regular" Milman position, thereby improving a result of G. Pisier. We establish a non-symmetric analogue of the strong Gaussian (B)-theorem which may be of independent interest.
Related
- Optimal MM* bounds for convex bodies
- Optimal stability of Pál's isominwidth inequality for ball convex bodies in planes of constant curvature
- The Petty Conjecture for Convex Bodies of Revolution
- Riemannian convex bodies with a geodesic segment on the boundary
- Continuity and monotonicity of weighted perimeters of convex bodies
BibTeX
@misc{paouris2026optimal,
title = {{Optimal mean width and metric entropy estimates for convex bodies}},
author = {Grigoris Paouris and Reese Pathak},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.29522},
eprint = {2607.29522},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.29522},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal mean width and metric entropy estimates for convex bodies
AU - Grigoris Paouris
AU - Reese Pathak
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2607.29522
UR - https://arxiv.org/abs/2607.29522
AB - We show that for any n ≥ 1 and any convex body K ⊂ R^n, there exists T ∈ SL(n) such that diam(TK) ≲ √n vr(K), and M^∗(T(K−x) ∩ rvr(K) B^n_2)≲ vr(K) √(log(e r^2)), for every x ∈ K and every r ≥ 1. Above, M^∗(·) denotes the spherical mean width and vr(·) denotes the volume radius. As a consequence, we establish for any convex body K ⊂ R^n that 1 ≤ inf_T ∈ SL(n) M^∗(TK)/vr(K) ≲ √(log(e n)). The estimates above are sharp, up to universal constants, as they are attained for the crosspolytope and any regular n-simplex. Up to universal constants, our results show that all quermassintegrals, and the logarithm of the covering numbers for all scales, simultaneously for K and the polar body K^∘, are maximized by the simplex and crosspolytope. Our proof makes use of Eldan's stochastic localization. To establish the results, we work with an extension of Bobkov's maximal Gaussian measure position to possibly non-symmetric convex bodies. Our results imply that this position is an optimal "regular" Milman position, thereby improving a result of G. Pisier. We establish a non-symmetric analogue of the strong Gaussian (B)-theorem which may be of independent interest.
ER -