On levels in arrangements of lines, segments, planes, and triangles
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Motion planning of a ball amid segments in three dimensions
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Nice point sets can have nasty Delaunay triangulations
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Selecting Points that are Heavily Covered by Pseudo-Circles, Spheres or Rectangles
Combinatorics, Probability and Computing
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A collection of geometric selection lemmas is proved, such as the following: For any set $P$ of $n$ points in three-dimensional space and any set ${\cal S}$ of $m$ spheres, where each sphere passes through a distinct point pair in $P$, there exists a point $x$, not necessarily in $P$, that is enclosed by $\Omega (m^2/(n^2 \log^6 {n^2 \over m}))$ of the spheres in ${\cal S}$. Similar results apply in arbitrary fixed dimensions, and for geometric bodies other than spheres. The results have applications in reducing the size of geometric structures, such as three-dimensional Delaunay triangulations and Gabriel graphs, by adding extra points to their defining sets.