Line Transversals to Unit Disks

Abstract. We show that if every three members of a finite disjoint family of unit disks in the plane have a line transversal, then there is a line transversal to all except at most 12 disks in the family. We derive an analogous result for translates of a general compact convex set, with the constant equal to 47.     

Line Transversals to Unit Disks

   Abstract. We show that if every three members of a finite disjoint family of unit disks in the plane have a line transversal, then there is a line transversal to all except at most 12 disks in the family. We derive an analogous result for translates of a general compact convex set, with the constant equal to 47.     

Polyhedral line transversals in space

Algorithms are developed for determining if a set of polyhedral objects inR ³ can be intersected by a common transversal (stabbing) line. It can be determined inO(n logn) time if a set ofn line segments in space has a line transversal, and such a transversal can be found in the same time bound. For a set of polyhedra with a total ofn vertices, we give anO(n ⁴ logn) algorithm for determining the existence of, and computing, a line transversal. Helly-type theorems for lines and segments are also given. In particular, it is shown that if every six of a set of lines in space are intersected by a common transversal, then the entire set has a common transversal.     

On the transversal Helly numbers of disjoint and overlapping disks

A family of disks is said to have the property T(k) if any k members of the family have a common line transversal. We call a family of unit diameter disks t-disjoint if the distances between the centers are greater than t. We consider for each natural number k≧ 3 the infimum t_k of the distances t for which any finite family of t-disjoint unit diameter disks with the property T(k) has a line transversal. We determine exact values of t₃ and t₄, and give general lower and upper bounds on the sequence t_k, showing that t_k = O(1/k) as k → ∞. In honour of Helge Tverberg’s seventieth birthday Received: 9 June 2005     

A gallai-type transversal problem in the plane

LetK be a family of compact convex sets in the plane. We show that if every three members ofK admit a common line transversal, then there exist four lines which together meet all the members ofK.     

Separating convex sets by straight lines

We answer some questions of Tverberg about separability properties of families of convex sets. In particular, we show that there is a family of infinitely many pairwise disjoint closed disks, no two of which can be separated from two others by a straight line. No such construction exists with equal disks. We also prove that every uncountable family of pairwise disjoint convex sets in the plane has two uncountable subfamilies that can be separated by a straight line.     

A Helly-Type Theorem for Line Transversals to Disjoint Unit Balls

Abstract. Let F be a family of disjoint unit balls in R ³ . We prove that there is a Helly-number n ₀ ≤ 46 , such that if every n ₀ members of F ( | F | ≥ n ₀ ) have a line transversal, then F has a line transversal. In order to prove this we prove that if the members of F can be ordered in a way such that every 12 members of F are met by a line consistent with the ordering, then F has a line transversal. The proof also uses the recent result on geometric permutations for disjoint unit balls by Katchalski, Suri, and Zhou.     

A Helly-Type Theorem for Line Transversals to Disjoint Unit Balls

   Abstract. Let F be a family of disjoint unit balls in R ³ . We prove that there is a Helly-number n ₀ ≤ 46 , such that if every n ₀ members of F ( | F | ≥ n ₀ ) have a line transversal, then F has a line transversal. In order to prove this we prove that if the members of F can be ordered in a way such that every 12 members of F are met by a line consistent with the ordering, then F has a line transversal. The proof also uses the recent result on geometric permutations for disjoint unit balls by Katchalski, Suri, and Zhou.     

A generalization of hadwiger's transversal theorem to intersecting sets

Given an ordered family of compact convex sets in the plane, if every three sets can be intersected by some directed line consistent with the ordering, then there exists a common transversal of the family. This generalizes Hadwiger's Transversal Theorem to families of compact convex sets which are not necessarily pairwise disjoint. If every six sets can be intersected by some directed line consistent with the ordering, then there exists a common transversal which is consistent with the ordering. If the family is pairwise disjoint and every four sets can be intersected by some directed line consistent with the ordering, then there exists a common transversal which is consistent with the ordering.     

Line Transversals in Large <Emphasis Type="Italic">T</Emphasis>(3)- and <Emphasis Type="Italic">T</Emphasis>(4)-Families of Congruent Discs

A family of closed discs is said to have property T(k) if to every subset of at most k discs there belongs a common line transversal. A family of discs is said to be d-disjoint, d≥1, if the mutual distance between the centers of the discs is larger than d. It is known that a d-disjoint T(3)-family ℱ of unit diameter discs has a line transversal if . Similarly, a d-disjoint T(4)-family has a line transversal if . Both results are sharp in d, i.e., they do not hold for smaller values of d. The main result of this paper is that while the above lower bounds on d cannot be relaxed in general, some reduction of d can be compensated by imposing a proper d-dependent lower bound on the size of the family in both cases.     

Allowable Interval Sequences and Line Transversals in the Plane

Given a family F of n pairwise disjoint compact convex sets in the plane with non-empty interiors, let T(k) denote the property that every subfamily of F of size k has a line transversal, and T the property that the entire family has a line transversal. We illustrate the applicability of allowable interval sequences to problems involving line transversals in the plane by proving two new results and generalizing three old ones. Two of the old results are Klee??s assertion that if F is totally separated then T(3) implies T, and the following variation of Hadwiger??s Transversal Theorem proved by Wenger and (independently) Tverberg: If F is ordered and each four sets of F have some transversal which respects the order on F, then there is a transversal for all of F which respects this order. The third old result (a consequence of an observation made by Kramer) and the first of the new results (which partially settles a 2008 conjecture of Eckhoff) deal with fractional transversals and share the following general form: If F has property T(k) and meets certain other conditions, then there exists a transversal of some m sets in F, with k<m<n. The second new result establishes a link between transversal properties and separation properties of certain families of convex sets.     

A colorful theorem on transversal lines to plane convex sets

We prove a colorful version of Hadwiger’s transversal line theorem: if a family of colored and numbered convex sets in the plane has the property that any three differently colored members have a transversal line that meet the sets consistently with the numbering, then there exists a color such that all the convex sets of that color have a transversal line. All authors are partially supported by CONACYT research grant 5040017.     

Proof of the Katchalski-Lewis Transversal Conjecture for T(3)-Families of Congruent Discs

A family of disjoint closed congruent discs is said to have property T(3) if to every triple of discs there exists a common line transversal. Katchalski and Lewis [10] proved the existence of a constant m_disc such that to every family of disjoint closed congruent discs with property T(3) a straight line can be found meeting all but at most m_disc of the members of the family. They conjectured that this is true even with m_disc = 2. On one hand Bezdek [1] proved m_disc ≥ 2 in 1991 and on the other hand Kaiser [9] showed m_disc ≤ 12 in a recent paper. The present work is devoted to proving this conjecture showing that m_disc ≤ 2.     

完全单半群及完全正则半群的逆断面

指出完全单半群Ｓ的任何一个Ｆ－类是逆断面,且为Ｑ－逆断面,而Ｓ的任何一个逆断面必是一个Ｆ－类,因而所有逆断面同构。并且给出完全正则半群的逆断面存在的充要条件。     

Total domination of graphs and small transversals of hypergraphs

The main result of this paper is that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n + 4m)/21 vertices. In the particular case n = m, the transversal has at most 3n/7 vertices, and this bound is sharp in the complement of the Fano plane. Chvátal and McDiarmid [5] proved that every 3-uniform hypergraph with n vertices and edges has a transversal of size n/2. Two direct corollaries of these results are that every graph with minimal degree at least 3 has total domination number at most n/2 and every graph with minimal degree at least 4 has total domination number at most 3n/7. These two bounds are sharp.     

A Helly-type theorem for higher-dimensional transversals

We prove that a collection of compact convex sets of bounded diameters in that is unbounded in k independent directions has a k-flat transversal for k<d if and only if every d+1 of the sets have a k-transversal. This result generalizes a theorem of Hadwiger(–Danzer–Grünbaum–Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d−1.     

No Helly Theorem for Stabbing Translates by Lines in R<Superscript>3</Superscript>

For each $n>2$ we construct a convex body $K\subset {\Bbb R}^3$ and a finite family ${\cal F}$ of disjoint translates of $K$ such that any $n-1$ members ${\cal F}$ admit a line transversal, but ${\cal F}$ has no line transversal.     

Line transversals to blown up closed balls

The number ?? _d (k) is defined as the minimum ???>?0 such that the following holds: For any finite family ${\mathcal {F}=\{B_1,B_2, \ldots , B_n\}}$ of closed balls in ${{\mathbb{R}}^d}$ such that every k elements of ${\mathcal {F}}$ have a common line transversal, the elements of the blown up family ${\lambda\mathcal {F}=\{\lambda B_1,\lambda B_2, \ldots , \lambda B_n\}}$ have a common line transversal. In this paper we show that ${\lambda_d(d+1)\leq4, \lambda_2(4)\leq 2\sqrt 2}$ and ??₂(3)?<?2.88.     

5-一致超图的全横贯

设H=(V,E)是以V为顶点集, E为(超)边集的超图. 如果H的每条边均含有k个顶点, 则称H是k-一致超图. 超图H的点子集T称为它的一个横贯, 如果T 与H 的每条边均相交. 超图H的全横贯是指它的一个横贯T, 并且T还满足如下性质: T中每个顶点均至少有一个邻点在T中. H 的全横贯数定义为H 的最小全横贯所含顶点的数目, 记作\tau_{t}(H). 对于整数k\geq 2, 令b_{k}=\sup_{H\in{\mathscr{H}}_{k}}\frac{\tau_{t}(H)}{n_{H}+m_{H}}, 其中n_H=|V|, m_H=|E|, {\mathscr{H}}_{k} 表示无孤立点和孤立边以及多重边的k-一致超图类. 最近, Bujt\'as和Henning等证明了如下结果: b_{2}=\frac{2}{5}, b_{3}=\frac{1}{3}, b_{4}=\frac{2}{7}; 当k\geq 5 时, 有b_{k}\leq \frac{2}{7}以及b_{6}\leq \frac{1}{4}; 当k\geq 7 时, b_{k}\leq \frac{2}{9}. 证明了对5-一致超图, b_{5}\leq \frac{4}{15}, 从而改进了当k=5 时b_k的上界.     

Union of linear spaces in infinite-dimensional projective spaces and holomorphic vector bundles

Let V be a localizing Banach space with an unconditional countable basis, X an equicodimensional transversal union of finite-codimensional linear subspaces of P(V) and E a holomorphic vector bundle of finite rank on X. Here we prove that Hⁱ(X,E) = 0 for every i > 0 and that E is isomorphic to a direct sum of line bundles ^OX(t).