Combinatorial complexity bounds for arrangements of curves and spheres
Discrete & Computational Geometry - Special issue on the complexity of arrangements
The number of different distances determined by a set of points in the Euclidean plane
Discrete & Computational Geometry
Incidences between points and circles in three and higher dimensions
Proceedings of the eighteenth annual symposium on Computational geometry
Lectures on Discrete Geometry
Distinct distances in homogeneous sets
Proceedings of the nineteenth annual symposium on Computational geometry
Crossing Numbers and Hard Erdös Problems in Discrete Geometry
Combinatorics, Probability and Computing
On the number of tetrahedra with minimum, unit, and distinct volumes in three-space
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
On the number of tetrahedra with minimum, unit, and distinct volumes in three-space
Combinatorics, Probability and Computing
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Improving an old result of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl, we show that the number of distinct distances determined by a set $P$ of $n$ points in three-dimensional space is $\Omega(n^{77/141-\varepsilon})=\Omega(n^{0.546})$, for any $\varepsilon0$. Moreover, there always exists a point $p\in P$ from which there are at least so many distinct distances to the remaining elements of $P$. The same result holds for points on the three-dimensional sphere. As a consequence, we obtain analogous results in higher dimensions.