Abstract
We study the lower bound for Koldobsky’s slicing inequality. We show that there exists a measure μ and a symmetric convex body \(K \subseteq \mathbb R^n\), such that for all \(\xi \in {{\mathbb S}^{n-1}}\) and all \(t\in \mathbb R,\)
Our bound is optimal, up to the value of the universal constant. It improves slightly upon the results of the first named author and Koldobsky, which included a doubly-logarithmic error. The proof is based on an efficient way of discretizing the unit sphere.
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Acknowledgements
The second named author is supported in part by the NSF CAREER DMS-1753260. The work was partially supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. The authors are grateful to Alexander Koldobsky for fruitful discussions and helpful comments. The authors are thankful to the anonymous referee for valuable suggestions.
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Klartag, B., Livshyts, G.V. (2020). The Lower Bound for Koldobsky’s Slicing Inequality via Random Rounding. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2266. Springer, Cham. https://doi.org/10.1007/978-3-030-46762-3_2
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DOI: https://doi.org/10.1007/978-3-030-46762-3_2
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