# 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} }

AbstractLetA be a family ofn pairwise disjoint compact convex sets inRd. Let
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## 48 Citations

### Improved Bounds for Geometric Permutations

- Mathematics, Computer Science2010 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 ScienceSCG '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

- Mathematics
- 2018

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 ScienceSODA '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

- MathematicsSODA '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 ScienceDiscret. 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

- MathematicsDiscret. 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)

- MathematicsDiscret. 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

- Mathematics
- 2016

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…

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