November 2022 Slicing p-balls reloaded: Stability, planar sections in 1
Giorgos Chasapis, Piotr Nayar, Tomasz Tkocz
Author Affiliations +
Ann. Probab. 50(6): 2344-2372 (November 2022). DOI: 10.1214/22-AOP1584


We show that the two-dimensional minimum-volume central section of the n-dimensional cross-polytope is attained by the regular 2n-gon. We establish stability-type results for hyperplane sections of p-balls in all the cases where the extremisers are known. Our methods are mainly probabilistic, exploring connections between negative moments of projections of random vectors uniformly distributed on convex bodies and volume of their sections.

Funding Statement

The second author was supported by the National Science Centre, Poland, grant 2018/31/D/ST1/01355.
The third author’s research supported in part by NSF Grant DMS-1955175.


We would like to thank Fedor Nazarov for helpful discussions and for sharing with us his proof of Theorem 1, as well as letting us include it in this paper. We are also indebted to the anonymous referee for many valuable comments which helped significantly improve the manuscript.


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Giorgos Chasapis. Piotr Nayar. Tomasz Tkocz. "Slicing p-balls reloaded: Stability, planar sections in 1." Ann. Probab. 50 (6) 2344 - 2372, November 2022.


Received: 1 September 2021; Revised: 1 February 2022; Published: November 2022
First available in Project Euclid: 23 October 2022

MathSciNet: MR4499279
zbMATH: 1506.52003
Digital Object Identifier: 10.1214/22-AOP1584

Primary: 52A40
Secondary: 52A20

Keywords: Convex bodies , negative moments , p-norm , stability , volume of sections

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 6 • November 2022
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