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Ordinary hyperspheres and spherical curves

Lin, Aaron and Swanepoel, Konrad ORCID: 0000-0002-1668-887X (2021) Ordinary hyperspheres and spherical curves. Advances in Geometry, 21 (1). 15 - 22. ISSN 1615-715X

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Identification Number: 10.1515/advgeom-2020-0031


An ordinary hypersphere of a set of points in real d-space, where no d + 1 points lie on a (d - 2)-sphere or a (d - 2)-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly d + 1 points of the set. Similarly, a (d + 2)-point hypersphere of such a set is one that contains exactly d + 2 points of the set. We find the minimum number of ordinary hyperspheres, solving the d-dimensional spherical analogue of the Dirac-Motzkin conjecture for d ≥ 3. We also find the maximum number of (d + 2)-point hyperspheres in even dimensions, solving the d-dimensional spherical analogue of the orchard problem for even d ≥ 4.

Item Type: Article
Official URL:
Additional Information: © 2021 Walter de Gruyter GmbH, Berlin/Boston.
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 24 Mar 2020 13:57
Last Modified: 01 Jan 2022 00:00

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