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Ordinary hyperspheres and spherical curves

Aaron Lin and Konrad Swanepoel
From the journal Advances in Geometry

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.

MSC 2010: 52B05

  1. Communicated by: J. Bamberg

References

[1] A. Bálintová, V. Bálint, On the number of circles determined by n points in the Euclidean plane. Acta Math. Hungar. 63 (1994), 283–289. MR1261471 Zbl 0796.51008Search in Google Scholar

[2] P. D. T. A. Elliott, On the number of circles determined by n points. Acta Math. Acad. Sci. Hungar. 18 (1967), 181–188. MR213939 Zbl 0163.14701Search in Google Scholar

[3] B. Green, T. Tao, On sets defining few ordinary lines. Discrete Comput. Geom. 50 (2013), 409–468. MR3090525 Zbl 1309.51002Search in Google Scholar

[4] J. Harris, Algebraic geometry Springer 1992. MR1182558 Zbl 0779.14001Search in Google Scholar

[5] K. Hulek, Elementary algebraic geometry American Mathematical Society 2003. MR1799530 Zbl 0957.14001Search in Google Scholar

[6] W. W. Johnson, Classification of plane curves with reference to inversion. Analyst 4 (1877), 42–47. JFM 09.0482.02Search in Google Scholar

[7] A. Lin, M. Makhul, H. N. Mojarrad, J. Schicho, K. Swanepoel, F. de Zeeuw, On sets defining few ordinary circles. Discrete Comput. Geom. 59 (2018), 59–87. MR3738336 Zbl 1384.52023Search in Google Scholar

[8] A. Lin, K. Swanepoel, On sets defining few ordinary hyperplanes. Discrete Anal. 2020, Article No. 4, 34 pp. MR4107320 Zbl 07263000Search in Google Scholar

[9] D. Pedoe, Geometry Dover Publications, New York 1988. MR1017034 Zbl 0716.51002Search in Google Scholar

[10] G. B. Purdy, J. W. Smith, Lines, circles, planes and spheres. Discrete Comput. Geom. 44 (2010), 860–882. MR2728037 Zbl 1227.05105Search in Google Scholar

[11] M. Reid, Undergraduate algebraic geometry volume 12 of London Mathematical Society Student Texts Cambridge Univ. Press 1988. MR982494 Zbl 0701.14001Search in Google Scholar

[12] J. G. Semple, L. Roth, Introduction to algebraic geometry Oxford Univ. Press 1985. MR814690 Zbl 0576.14001Search in Google Scholar

[13] R. Zhang, On the number of ordinary circles determined by n points. Discrete Comput. Geom. 46 (2011), 205–211. MR2812505 Zbl 1222.51013Search in Google Scholar

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

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