Ordinary hyperspheres and spherical curves

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

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Abstract

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:	https://www.degruyter.com/view/journals/advg/advg-...
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:	20 Sep 2025 01:50
URI:	http://eprintstest.lse.ac.uk/id/eprint/103821

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