## Abstract

Let P be a set of n points in ℝ^{3}, and let k ≤ n be an integer. A sphere σ is k-rich with respect to P if |σ ∩ P| ≥ k, and is η-non-degenerate, for a fixed fraction 0 < η < 1, if no circle γ ⊂, σ contains more than η|σ ∩ P| points of P. We improve the previous bound given in [1] on the number of k-rich η-non-degenerate spheres in 3-space with respect to any set of n points in ℝ^{3}, from O(n^{4}/k^{5} + n^{3}/k ^{3}), which holds for all 0 < η < 1/2, to O*(n ^{4}/k^{11/2} + n^{2}/k^{2}), which holds for all 0 < η < 1 (in both bounds, the constants of proportionality depend on η). The new bound implies the improved upper bound O*(n ^{58/27}) ≈ O(n^{2.1482}) on the number of mutually similar triangles spanned by n points in ℝ^{3}; the previous bound was O(n^{13/6}) ≈ O(n^{2.1667}) [1].

Original language | English |
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Pages (from-to) | 503-512 |

Number of pages | 10 |

Journal | Combinatorics Probability and Computing |

Volume | 20 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2011 |