Freyer, A., & Henk, M. (2024). Polynomial bounds in Koldobsky’s discrete slicing problem. Proceedings of the American Mathematical Society, 152(7), 3063–3074. https://doi.org/10.1090/proc/16753
E104-06 - Forschungsbereich Konvexe und Diskrete Geometrie
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Journal:
Proceedings of the American Mathematical Society
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ISSN:
0002-9939
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Date (published):
1-Jul-2024
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Number of Pages:
12
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Publisher:
AMER MATHEMATICAL SOC
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Peer reviewed:
Yes
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Keywords:
lattice points; slicing problem; bounds
en
Abstract:
In 2013, Koldobsky posed the problem to find a constant dn, depending only on the dimension n, such that for any origin-symmetric convex body K ⊂ Rn there exists an (n - 1)-dimensional linear subspace H ⊂ Rn with |K ∩ Zn| ≤ dn |K ∩ H ∩ Zn| vol(K) n1 . In this article we show that dn is bounded from above by c n2 ω(n)/log(n), where c is an absolute constant and ω(n) is the flatness constant. Due to the recent best known upper bound on ω(n) we get a c n3 log(n)2 bound on dn. This improves on former bounds which were exponential in the dimension.
