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.

中文翻译：

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普通超球面和球面曲线

摘要 实 d 空间中一组点的普通超球面，其中没有 d + 1 个点位于 (d - 2) - 球面或 (d - 2) - 平面上，是超球面（包括退化情况一个超平面）恰好包含该集合的 d + 1 个点。类似地，此类集合的 (d + 2) 点超球面恰好包含该集合的 d + 2 个点。我们找到了最小数量的普通超球面，解决了狄拉克-莫茨金猜想的 d 维球面类似物对于 d ⩾ 3。我们还找到了偶数维中的 (d + 2) 点超球面的最大数量，解决了 d-对于偶数 d ⩾ 4，果园问题的维球面模拟。