Similar simplices in a d-dimensional point set

Abstract

We consider the problem of bounding the maximum possible number f k,d (n) of ksimplices that are spanned by a set of n points in R d and are similar to a given simplex. We first show that f 2,3 (n) = O(n 13/6 ), and then tackle the general case, and show

Improved bounds for incidences between points and circles

“…It remains to bound I(P (1) p , C0). For this, we again use an analysis similar to the one in Theorem 1.…”

“…. , t, set P (1) p,i = Pp ∩ Z(fi) and mp,i = |P (1) p,i |. Let np,i denote the number of circles of C0 that are fully contained in Z(fi).…”

“…When the circles have arbitrary radii, the current best bound for any dimension d ≥ 2 (originally established for the planar case in [3,8,28], and later extended to higher dimensions by Aronov et al [7]) is I(P, C) = O * ( m 2/3 n 2/3 + m 6/11 n 9/11 + m + n . (1) Here in fact the O * (·) notation only hides polylogarithmic factors; the precise best known upper bound is O(m 2/3 n 2/3 + m 6/11 n 9/11 log 2/11 (m 3 /n) + m + n) [28].…”

“…(1) In the planar case, the best known lower bound for the number of incidences between points and circles is Ω * (m 2/3 n 2/3 + m + n) (e.g., see [33]) 2 . Theorem 1.1 implies that for certain ranges of m, n, and q, a smaller upper bound holds in R 3 .…”

“…(4) Interestingly, the "threshold" value m = Θ(n 3/2 ) where a quantitative change in the bound takes place (as noted 1 There is no real difference between the cases of coplanarity and cosphericality of the points and circles, since the latter case can be reduced to the former (and vice versa) by means of the stereographic projection. 2 It is in fact larger than the explicit expression by a fractional logarithmic factor.…”