Abstract
The maximum of the Banach–Mazur distance \(d_{BM}^M(X,\ell_\infty^n)\), where \(X\) ranges over the set of all \(n\)-dimensional real Banach spaces, is difficult to compute. In fact, it is even not easy to find the maximum of \(d_{BM}^M(\ell_p^n,\ell_\infty^n)\) over all \(p\in [1,\infty]\). We prove that \(d_{BM}^M(\ell_p^3,\ell_\infty^3)\leq 9/5\) for all \(p\in[1,\infty]\). As an application, the following result related to Borsuk’s partition problem in Banach spaces is obtained: any subset \(A\) of \(\ell_p^3\) having diameter \(1\) is a union of \(8\) subsets of \(A\) whose diameters are at most \(0.9\).
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Funding
The authors are supported by the National Natural Science Foundation of China (grant numbers 12071444 and 12001500), the Fundamental Research Program of Shanxi Province (grant numbers 201901D111141, 202103021223191, and 202303021221116), the 19th Graduate Science and Technology Project of North University of China (grant number 20231944), and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (grant number 2020L0290).
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Zhang, L., Meng, L. & Wu, S. Banach–Mazur Distance from \(\ell_p^3\) to \(\ell_\infty^3\). Math Notes 114, 1045–1051 (2023). https://doi.org/10.1134/S0001434623110354
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DOI: https://doi.org/10.1134/S0001434623110354