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On the Partition of Plane Sets Into 6 Subsets of Small Diameter

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In 1956, H. Lenz introduced a problem to find members of the sequence

$${d}_{n}=\underset{\Phi }{\mathrm{inf}}\left\{x\in {\mathbb{R}}^{+}:\Phi \subset {\Phi }_{1}\cup {\Phi }_{2}\cup \cdots \cup {\Phi }_{n},\mathrm{ diam }{\Phi }_{i}\le x for all i\right\}$$

where the infimum is taken over all sets Φ of unit diameter. In the present paper, an upper bound for d6 is improved to 0.53432….

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Authors and Affiliations

  1. Uniwersytet Jagielloński (Jagiellonian university), Kraków, Polska

    V. O. Koval

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  1. V. O. Koval
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Correspondence to V. O. Koval.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 497, 2020, pp. 100–123.

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Koval, V.O. On the Partition of Plane Sets Into 6 Subsets of Small Diameter. J Math Sci 275, 177–194 (2023). https://doi.org/10.1007/s10958-023-06670-0

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