## On Lines and Joints (2009)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Kaplan09onlines,

author = {Haim Kaplan and Micha Sharir and Eugenii Shustin},

title = {On Lines and Joints },

year = {2009}

}

### OpenURL

### Abstract

Let L be a set of n lines in R d, for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1)). For d = 3, this is a considerable simplification of the orignal algebraic proof of Guth and Katz [9], and of the follow-up simpler proof of Elekes et al. [6]. Some extensions, e.g., to the case of joints of algebraic curves, are also presented.

### Citations

346 |
Using Algebraic Geometry
- Cox, Little, et al.
- 2005
(Show Context)
Citation Context ...s at least mnj/(2n) surviving joints, and this quantity is larger than bnj, by the initial assumption as in the preceding proof. Since p vanishes at all these points, it follows from Bézout’s theorem =-=[4]-=- that p ≡ 0 on each Fj. The proof then continues exactly as before, showing that all partial derivatives of p, of any order, vanish on all the surviving curves Fj, which yields a contradiction, as abo... |

62 | Recent work connected with the Kakeya problem. Prospects in mathematics (Princeton, NJ
- Wolff
- 1996
(Show Context)
Citation Context ...inatorial geometry, including forbidden subgraphs in extremal graph theory, space decomposition techniques, and some basic results in the geometry of lines in space (e.g., Plücker coordinates). Wolff =-=[12]-=- observed a connection between the problem of counting joints to the Kakeya problem. Bennett et al. [1] exploited this connection and proved an upper bound on the number of so-called θ-transverse join... |

25 | On the size of Kakeya sets in finite fields
- Dvir
(Show Context)
Citation Context ...t the correct upper bound on the number of joints (in three dimensions) is O(n 3/2 ), matching the lower bound just noted. In a rather dramatic recent development, building on a recent result of Dvir =-=[5]-=- for a variant of the problem for finite fields, Guth and Katz [9] have settled the conjecture in the affirmative, showing that the number of joints (in three dimensions) is indeed O(n 3/2 ). Their pr... |

18 | On the multilinear restriction and Kakeya conjectures
- Bennett, Carbery, et al.
(Show Context)
Citation Context ...es, and some basic results in the geometry of lines in space (e.g., Plücker coordinates). Wolff [12] observed a connection between the problem of counting joints to the Kakeya problem. Bennett et al. =-=[1]-=- exploited this connection and proved an upper bound on the number of so-called θ-transverse joints in R 3 , namely, joints incident to at least one triple ∗ Work on this paper has been partly support... |

18 | Counting and cutting cycles of lines and rods in space
- Chazelle, Edelsbrunner, et al.
- 1990
(Show Context)
Citation Context ...aximum possible number of joints in a set of n lines in R d is Θ(n d/(d−1) ). Background. The problem of bounding the number of joints, for the 3-dimensional case, has been around for almost 20 years =-=[3, 7, 11]-=- (see also [2, Chapter 7.1, Problem 4]), and, until very recently, the best known upper bound, established by Sharir and Feldman [7], was O(n 1.6232 ). The proof techniques were rather complicated, in... |

14 | Algebraic methods in discrete analogs of the Kakeya problem
- Guth, Katz
(Show Context)
Citation Context ...e algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1) ). For d = 3, this is a considerable simplification of the orignal algebraic proof of Guth and Katz =-=[9]-=-, and of the follow-up simpler proof of Elekes et al. [6]. Some extensions, e.g., to the case of joints of algebraic curves, are also presented. Let L be a set of n lines in R d , for d ≥ 3. A joint o... |

12 | Introduction to Singularities and Deformations - Greuel, Lossen, et al. - 2006 |

10 |
On joints in arrangements of lines in space and related problems
- Sharir
- 1994
(Show Context)
Citation Context ...aximum possible number of joints in a set of n lines in R d is Θ(n d/(d−1) ). Background. The problem of bounding the number of joints, for the 3-dimensional case, has been around for almost 20 years =-=[3, 7, 11]-=- (see also [2, Chapter 7.1, Problem 4]), and, until very recently, the best known upper bound, established by Sharir and Feldman [7], was O(n 1.6232 ). The proof techniques were rather complicated, in... |

9 | An improved bound for joints in arrangements of lines in space. Discrete Comput - Feldman, Sharir |

3 |
On lines, joints, and incidences in three dimensions, manuscript
- Elekes, Sharir
- 2009
(Show Context)
Citation Context ...sible number of joints of L is Θ(n d/(d−1) ). For d = 3, this is a considerable simplification of the orignal algebraic proof of Guth and Katz [9], and of the follow-up simpler proof of Elekes et al. =-=[6]-=-. Some extensions, e.g., to the case of joints of algebraic curves, are also presented. Let L be a set of n lines in R d , for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all... |

3 |
The joints problem
- Quilodrán
(Show Context)
Citation Context ...o at least d curves whose tangents at a do not all lie in a common hyperplane. In a rather surprising turn of events, the same results were obtained independently and simultaneously 1 by R. Quilodrán =-=[10]-=-. The main ingredient of the proof is the same in both papers, but we believe that the peripheral part of the analysis is simpler in our paper. We also note that this paper does not subsume the previo... |