Abstract
In 1993, Kahn and Kalai famously constructed a sequence of finite sets in d-dimensional Euclidean spaces that cannot be partitioned into less than \({{(1.203 \ldots + o(1))}^{{\sqrt d }}}\) parts of smaller diameter. Their method works not only for the Euclidean, but for all \({{\ell }_{p}}\)-spaces as well. In this short note, we observe that the larger the value of p, the stronger this construction becomes.
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ACKNOWLEDGMENTS
We thank Olga Kostina for the helpful discussion. The first author is supported by the grant NSh-775.2022.1.1. The second author is supported by ERC Advanced Grant ‘GeoScape’ No. 882971.
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This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
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Raigorodskii, A.M., Sagdeev, A. A Note on Borsuk’s Problem in Minkowski Spaces. Dokl. Math. 109, 80–83 (2024). https://doi.org/10.1134/S1064562424701849
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DOI: https://doi.org/10.1134/S1064562424701849