Abstract
We show that the two-dimensional minimum-volume central section of the n-dimensional cross-polytope is attained by the regular -gon. We establish stability-type results for hyperplane sections of -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.
Acknowledgments
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.
Citation
Giorgos Chasapis. Piotr Nayar. Tomasz Tkocz. "Slicing -balls reloaded: Stability, planar sections in ." Ann. Probab. 50 (6) 2344 - 2372, November 2022. https://doi.org/10.1214/22-AOP1584
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