The colored Hadwiger transversal theorem in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

Hadwiger’s transversal theorem gives necessary and suffcient conditions for a family of convex sets in the plane to have a line transversal. A higher dimensional version was obtained by Goodman, Pollack and Wenger, and recently a colorful version appeared due to Arocha, Bracho and Montejano. We show that it is possible to combine both results to obtain a colored version of Hadwiger's theorem in higher dimensions. The proofs differ from the previous ones and use a variant of the Borsuk-Ulam theorem. To be precise, we prove the following. Let F be a family of convex sets in ? ^d in bijection with a set P of points in ? ^d?1. Assume that there is a coloring of F with suffciently many colors such that any colorful Radon partition of points in P corresponds to a colorful Radon partition of sets in F. Then some monochromatic subfamily of F has a hyperplane transversal.     

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.     

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.     

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

A geometric permutation induced by a transversal line of a finite family of disjoint convex sets in ℝ^d is the order in which the transversal meets the members of the family. It is known that the maximal number of geometric permutations in families of n disjoint translates of a convex set in ℝ³ is 3. We prove that for d ≥ 3 the maximal number of geometric permutations for such families in ℝ^d is Ω(n).     

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.     

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.     

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.     

Forbidden Families of Geometric Permutations in ℝd

A geometric permutation induced by a transversal line of a finite family ℱ of disjoint convex sets in ℝ^d is the order in which the transversal meets the members of the family. We prove that for each natural k, each family of k permutations is realizable (as a family of geometric permutations of some ℱ) in ℝ^d for d ≥ 2k – 1, but there is a family of k permutations which is non-realizable in ℝ^d for d ≤ 2k – 2.     

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.     

Flat transversals to flats and convex sets of a fixed dimension

Helly and Hadwiger type theorems for transversal m-flats to families of flats and, respectively, convex sets of dimension n are proved in the case of general position. The proofs rely on Helly type theorems for “linear partitions” and “convex partitions,” so that a general theory of Helly numbers is also developed.     

List multicolorings of graphs with measurable sets

In an ordinary list multicoloring of a graph, the vertices are “colored” with subsets of pre‐assigned finite sets (called “lists”) in such a way that adjacent vertices are colored with disjoint sets. Here we consider the analog of such colorings in which the lists are measurable sets from an arbitrary atomless, semifinite measure space, and the color sets are to have prescribed measures rather than prescribed cardinalities. We adapt a proof technique of Bollobás and Varopoulos to prove an analog of one of the major theorems about ordinary list multicolorings, a generalization and extension of Hall's marriage theorem, due to Cropper, Gyárfás, and Lehel. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 179–193, 2007     

On the calculation of extended max and min operations between convex fuzzy sets of the real line

In this paper we will identify the sets of so-called sub- and pseudo-highest intersection points of convex fuzzy sets of the real line and will explore their properties. Based on the properties of these sets, an algorithm for calculating extended max and min operations between two or more convex fuzzy sets of the real line with general membership functions, not necessarily continuous, is proposed.     

Rational convexity of non-generic immersed Lagrangian submanifolds

We prove that an immersed Lagrangian submanifold in C ⁿ with quadratic self-tangencies is rationally convex. This generalizes former results for the embedded and the immersed transversal cases.     

A priori degeneracy of one-dimensional rotation sets for periodic point free torus maps

Diffeomorphisms of the two torus that are isotopic to the identity have rotation sets that are convex compact subsets of the plane. We show that certain line segments (including all rationally sloped segments with no rational points) cannot be realized as a rotation set.

     

Convex and strongly convex fuzzy sets

Properties of the support and of the core of convex and strongly convex fuzzy sets are considered. The convex and strongly convex fuzzy sets in the real line are characterized by means of the piece-wise monotonic functions.     

Efficient solutions in generalized linear vector optimization

This paper establishes several new facts on generalized polyhedral convex sets and shows how they can be used in vector optimization. Among other things, a scalarization formula for the efficient solution sets of generalized linear vector optimization problems is obtained. We also prove that the efficient solution set of a generalized linear vector optimization problem in a locally convex Hausdorff topological vector space is the union of finitely many generalized polyhedral convex sets and it is connected by line segments.     

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.     

Proof of grünbaum's conjecture on common transversals for translates

In 1958 B. Grünbaum made a conjecture concerning families of disjoint translates of a compact convex set in the plane: if such a family consists of at least five sets, and if any five of these sets are met by a common line, then some line meets all sets of the family. This paper gives a proof of the conjecture.     

Okounkov bodies and Seshadri constants

Okounkov bodies, which are closed convex sets defined for big line bundles, have rich information on the line bundles. On the other hand, Seshadri constants are invariants which measure the positivity of line bundles. In this paper, we prove that Okounkov bodies give lower bounds of Seshadri constants.