Corpus ID: 221655547

Empty axis-parallel boxes

@article{Bukh2020EmptyAB,
  title={Empty axis-parallel boxes},
  author={B. Bukh and Ting-Wei Chao},
  journal={ArXiv},
  year={2020},
  volume={abs/2009.05820}
}
We show that, for every set of $n$ points in the $d$-dimensional unit cube, there is an empty axis-parallel box of volume at least $\Omega(d/n)$ as $n\to\infty$ and $d$ is fixed. In the opposite direction, we give a construction without an empty axis-parallel box of volume $O(d^2\log d/n)$. These improve on the previous best bounds of $\Omega(\log d/n)$ and $O(2^{7d}/n)$ respectively. 
Piercing All Translates of a Set of Axis-Parallel Rectangles

References

SHOWING 1-10 OF 15 REFERENCES
On the size of the largest empty box amidst a point set
A note on minimal dispersion of point sets in the unit cube
On the Largest Empty Axis-Parallel Box Amidst n Points
An upper bound on the minimal dispersion
A lower bound for the dispersion on the torus
  • Mario Ullrich
  • Mathematics, Computer Science
  • Math. Comput. Simul.
  • 2018
Improved dispersion bounds for modified Fibonacci lattices
Computational Geometry Column 60
Tractability of the Approximation of High-Dimensional Rank One Tensors
Recovery algorithms for high-dimensional rank one tensors
...
1
2
...