A problem of Leo Moser about repeated distances on the sphere
American Mathematical Monthly
Combinatorial complexity bounds for arrangements of curves and spheres
Discrete & Computational Geometry - Special issue on the complexity of arrangements
On the zone of a surface in a hyperplane arrangement
Discrete & Computational Geometry
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Point set pattern matching in 3-D
Pattern Recognition Letters
Computing Many Faces in Arrangements of Lines and Segments
SIAM Journal on Computing
Crossing Numbers and Hard Erdös Problems in Discrete Geometry
Combinatorics, Probability and Computing
Incidences between points and circles in three and higher dimensions
Proceedings of the eighteenth annual symposium on Computational geometry
| Hi-index | 0.00 |
We derive improved bounds on the number of k-dimensional simplices spa nned by a set of n points in $\reals^d$ that are congruent to a given $k$-simplex, for $k\le d-1$. Let $f_k^{(d)} (n)$ be the maximum number of $k$-simplices spanned by a set of $n$ points in $\reals^d$ that are congruent to a given $k$-simplex. We prove that $f_2^{(3)}(n) = O(n^{5/3}\cdot 2^{O(\alpha^2(n))})$, $f_2^{(4)} (n) = O(n^{2+\eps})$, $f_2^{(5)} (n) = \Theta(n^{7/3})$, and $f_3^{(4)} (n) = O(n^{9/4+\eps})$. We also derive a recurrence to bound $f_k^{(d)} (n)$ for arbitrary values of $k$ and $d$, and use it to derive the bound $f_k^{(d)} (n) = O(n^{d/2})$ for $d \le 7$ and $k \le d-2$. Following Erd{\H o}s and Purdy, we conjecture that this bound holds for larger values of $d$ as well, and for $k\le d-2$.