Incidences between points and circles in three and higher dimensions

@inproceedings{Aronov2002IncidencesBP,
  title={Incidences between points and circles in three and higher dimensions},
  author={Boris Aronov and Vladlen Koltun and Micha Sharir},
  booktitle={SCG '02},
  year={2002}
}
(MATH) We show that the number of incidences between <i>m</i> distinct points and <i>n</i> distinct circles in $\reals^3$ is <i>O</i>(<i>m</i> <sup>4/7</sup> <i>n</i> <sup>17/21</sup>+<i>m</i> <sup>2/3</sup> <i>n</i> <sup>2/3</sup>+<i>m</i>+<i>n</i>); the bound is optimal for <i>m n</i> <sup>3/2</sup>. This result extends recent work on point-circle incidences in the plane, but its proof requires a different analysis. The bound improves upon a previous bound, noted by Akutsu et al. [2] and by… 
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References

SHOWING 1-10 OF 32 REFERENCES
Point-line incidences in space
TLDR
These bounds are smaller than the tight Szemerédi-Trotter bound for point-line incidences in $\reals^2$, unless both bounds are linear.
Lenses in arrangements of pseudo-circles and their applications
TLDR
A linear upper bound on the number of empty lenses in an arrangement of <i>n</i> pseudo-circles is established with the property that any two curves intersect precisely twice, and improved bounds for the number and complexity of incidences, the complexity of a single level, and thecomplexity of many faces are obtained.
Combinatorial complexity bounds for arrangements of curves and spheres
TLDR
Upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike are presented and it is proved that the maximum number of edges boundingm cells in an arrangement ofn lines is Θ(m2/3n 2/3 +n), and that it isO(m3/2β(m) forn unit-circles.
Improved Bounds for Planar k -Sets and Related Problems
  • T. Dey
  • Mathematics
    Discret. Comput. Geom.
  • 1998
TLDR
This is the first considerable improvement on this bound after its early solution approximately 27 years ago and applies to improve the current bounds on the combinatorial complexities of k -levels in the arrangement of line segments, convex polygons in the union of n lines, parametric minimum spanning trees, and parametric matroids in general.
Extremal problems in discrete geometry
TLDR
Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points.
On Levels in Arrangements of Curves
TLDR
The first subquadratic upper bound is given (roughly O( nk^ 1-1/(9· 2s-3) ) ) for curves that are graphs of polynomial functions of an arbitrary fixed degree s .
Lenses in arrangements of pseudo-circles and their applications
TLDR
It is shown that any collection of n pseudo-circles can be cut into $\bx$ arcs so that any two intersect at most once, provided that the given pseudo- circles are x-monotone and admit an algebraic representation by three real parameters.
Towards a Theory of Geometric Graphs
On the complexity of the linkage reconfiguration problem by H. Alt, C. Knauer, G. Rote, and S. Whitesides Falconer conjecture, spherical averages and discrete analogs by G. Arutyunyants and A.…
How to Cut Pseudoparabolas into Segments
TLDR
This work investigates how to cut pseudoparabolas into the minimum number of curve segments so that each pair of segments intersect at most once, and gives an Ω( n4/3 ) lower bound and O(n5/3) upper bound on the number of cuts.
Crossing Numbers and Hard Erd} os Problems in Discrete Geometry
We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the…
...
...