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Ordinary hyperspheres and spherical curves

Aaron Lin and Konrad Swanepoel
From the journal Advances in Geometry


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.

MSC 2010: 52B05

  1. Communicated by: J. Bamberg


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Received: 2019-05-23
Revised: 2020-01-29
Published Online: 2021-01-22
Published in Print: 2021-01-27

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