# Upper bounds on geometric permutations for convex sets

@article{Wenger1990UpperBO,
title={Upper bounds on geometric permutations for convex sets},
author={Rephael Wenger},
journal={Discrete \& Computational Geometry},
year={1990},
volume={5},
pages={27-33}
}
• R. Wenger
• Published 3 January 1990
• Mathematics
• Discrete & Computational Geometry

### Improved Bounds for Geometric Permutations

• Mathematics, Computer Science
2010 IEEE 51st Annual Symposium on Foundations of Computer Science
• 2010
The number of geometric permutations of an arbitrary collection of $n$ pair wise disjoint convex sets in $\mathbb{R}^d$ is O(n^{2d-3}\log n)$, improving Wenger's 20 years old bound of O (n 2d-2)$.

### A tight bound on the number of geometric permutations of convex fat objects in {\huge $\mathbf{\reals^d}$}

• Mathematics, Computer Science
SCG '01
• 2001
We show that the maximum number of geometric permutations of a set of $n$ pairwise-disjoint convex and fat objects in $\reals^d$ is $O(n^{d-1})$. This generalizes the bound of $\Theta (n^{d-1})$

### On K-Sets in Arrangements of Curves and Surfaces

We extend the notion ofk-sets and (≤k)-sets (see [3], [12], and [19]) to arrangements of curves and surfaces. In the case of curves in the plane, we assume that each curve is simple and separates the

### Line Transversals of Convex Polyhedra in $\reals^3$

• Mathematics
• 2008
The new bounds on the complexity (and construction cost) of $\T$ improve upon the previously best known bounds, which are nearly cubic in $n$.

### Geometric permutations of high dimensional spheres

• Mathematics, Computer Science
SODA '01
• 2001
The maximum number of geometric permutations, induced by line transversals to a set of pairwise disjoint congruent spheres in R<sup>d</sup></i>, is no more than 4 when <i>n</i> is sufficiently large, achieving the best known upper bound for this problem.

### Shape sensitive geometric permutations

• Mathematics
SODA '01
• 2001
We prove that a set of <i>n</i> unit balls in <i>R<sup>d</sup></i> admits at most <i>four</i> distinct geometric permutations, or line transversals, thus settling a long-standing conjecture in

### A Tight Bound on the Number of Geometric Permutations of Convex Fat Objects in Rd

• Mathematics, Computer Science
Discret. Comput. Geom.
• 2001
We show that the maximum number of geometric permutations of a set of n pairwise-disjoint convex and fat objects in Rd is O(nd-1) . This generalizes the bound of Θ (nd-1) obtained by Smorodinsky et

### Onk-sets in arrangements of curves and surfaces

• M. Sharir
• Mathematics
Discret. Comput. Geom.
• 1991
Borders that relate the maximum size of the (≤k)-set to the maximum sizes of a 0-set of a sample of the curves are obtained and various applications of these results are presented to arrangements of segments and curves, high-order Voronoi diagrams, partial stabbing of disjoint convex sets in the plane, and more.

### The Maximal Number of Geometric Permutations for n Disjoint Translates of a Convex Set in ℝ Is Ω(n)

• Mathematics
Discret. Comput. Geom.
• 2006
It is proved that for d ≥ 3 the maximal number of geometric permutations for such families in ℝd is Ω(n), which is the order in which the transversal meets the members of the family.

### HELLY-TYPE THEOREMS AND GEOMETRIC TRANSVERSALS

INTRODUCTION Let F be a family of convex sets in R. A geometric transversal is an affine subspace that intersects every member of F . More specifically, for a given integer 0 ≤ k < d, a k-dimensional

## References

SHOWING 1-9 OF 9 REFERENCES

### Geometric permutations and common transversals

• Mathematics
Discret. Comput. Geom.
• 1986
To study how many essentially different common transversals a family of convex sets on the plane can have, this work considers the case where the family consists of pairwise disjoint translates of a single convex set.

### A conjecture of Grünbaum on common transversals.

Demonstration d'une conjecture (faible) de Grunbaum sur l'existence d'une transversale commune a une famille finie d'ensembles convexes

### Computational geometry: an introduction

• Physics
• 1985
This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.

• New York,
• 1967