Finite Family Of Convex Sets Research Articles

A Family of Convex Sets in the Plane Satisfying the (4, 3)-Property can be Pierced by Nine Points

We prove that every finite family of convex sets in the plane satisfying the (4, 3)-property can be pierced by nine points. This improves the bound of 13 proved by Kleitman et al. (Combinatorica 21(2), 221–232 (2001)).

Eckhoff's Problem on Convex Sets in the Plane

Eckhoff proposed a combinatorial version of the classical Hadwiger–Debrunner $(p,q)$-problems as follows. Let ${\cal F}$ be a finite family of convex sets in the plane and let $m\geqslant 1$ be an integer. If among every ${m+2\choose 2}$ members of ${\cal F}$ all but at most $m-1$ members have a common point, then there is a common point for all but at most $m-1$ members of ${\cal F}$. The claim is an extension of Helly's theorem ($m=1$). The case $m=2$ was verified by Nadler and by Perles. Here we show that Eckhoff 's conjecture follows from an old conjecture due to Szemerédi and Petruska concerning $3$-uniform hypergraphs. This conjecture is still open in general; its solution for a few special cases answers Eckhoff's problem for $m=3,4$. A new proof for the case $m=2$ is also presented.

A new lower bound on Hadwiger-Debrunner numbers in the plane

A family of sets \({\cal F}\) is said to satisfy the (p, q) property if among any p sets in \({\cal F}\) some q have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any p ≥ q ≥ d + 1 there exists a minimum integer c = HDd(p, q), such that any finite family of convex sets in ℝd that satisfies the (p, q) property can be pierced by at most c points. In a celebrated result from 1992, Alon and Kleitman proved the conjecture. However, obtaining sharp bounds on HDd(p, q), known as ‘the Hadwiger-Debrunner numbers’, is still a major open problem in discrete and computational geometry.The best known upper bound on the Hadwiger-Debrunner numbers in the plane is \(O({p^{(1.5 + \delta)(1 + {1 \over {q - 2}})}})\) (for any δ > 0 and p ≥ q ≥ q0(δ)), obtained by combining the results of Keller, Smorodinsky and Tardos (2017) and of Rubin (2018). The best lower bound is \(H\,{D_2}(p,q) = \Omega \left({{p \over q}\log \left({{p \over q}} \right)} \right)\), obtained by Bukh, Matoušek and Nivasch more than 10 years ago.In this paper we improve the lower bound significantly by showing that HD2(p, q) ≥ p1+Ω(1/q). Furthermore, the bound is obtained by a family of lines and is tight for all families that have a bounded VC-dimension. Unlike previous bounds on the Hadwiger-Debrunner numbers, which mainly used the weak epsilon-net theorem, our bound stems from a surprising connection of the (p, q) problem to an old problem of Erdős on points in general position in the plane. We use a novel construction for Erdős’ problem, obtained recently by Balogh and Solymosi using the hypergraph container method, to get the lower bound on HD2(p, 3). We then generalize the bound to HD2(p, q) for q ≥ 3.

Quantitative combinatorial geometry for concave functions

We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in Rd has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a “witness set” which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and H-convex sets. Our results show that several new optimization problems can be solved with algorithms for LP-type problems. We obtain colorful and fractional variants of all our Helly-type theorems.

A Mélange of Diameter Helly-Type Theorems

A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in $\mathbb{R}^d$ given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove fractional and colorful versions of a long-standing conjecture by Bárány, Katchalski, and Pach. We also show that a Minkowski norm admits an exact Helly-type theorem for diameter if and only if its unit ball is a polytope and prove a colorful version for those that do. Finally, we prove Helly-type theorems for the property of “containing $k$ colinear integer points.”

Distributed Stochastic Approximation with Local Projections

We propose a distributed version of a stochastic approximation scheme constrained to remain in the intersection of a finite family of convex sets. The projection to the intersection of these sets i...

Convergence Analysis of Processes with Valiant Projection Operators in Hilbert Space

Convex feasibility problems require to find a point in the intersection of a finite family of convex sets. We propose to solve such problems by performing set-enlargements and applying a new kind of projection operators called valiant projectors. A valiant projector onto a convex set implements a special relaxation strategy, proposed by Goffin in 1971, that dictates the move toward the projection according to the distance from the set. Contrary to past realizations of this strategy, our valiant projection operator implements the strategy in a continuous fashion. We study properties of valiant projectors and prove convergence of our new valiant projections method. These results include as a special case and extend the 1985 automatic relaxation method of Censor.

Tight bounds on discrete quantitative Helly numbers

Let S be a discrete subset of Rn and define c(S,k) as the smallest number with the property that if a finite family of convex sets has exactly k points of S in common, then at most c(S,k) convex sets in this family already have exactly k points of S in common. For S=Zn, this number repeatedly appeared in different contexts as, for instance, optimization and geometry of numbers and, very recently, for general sets S, in the context of Helly and Tverberg theorems in De Loera et al. (2015).In this work, we give a useful description of c(S,k) in terms of polytopes with vertices in S. Starting with this description, we answer several fundamental questions about c(S,k). We provide the general upper bound c(S,k)≤⌊(k+1)/2⌋(c(S,0)−2)+c(S,0) for every discrete S. For the integer lattice S=Zn, employing techniques from the geometry of numbers, we solve the question on the asymptotic behavior by proving the estimate c(Zn,k)=Θ(k(n−1)/(n+1)) for every fixed n, and we compute the exact values of c(Zn,k) for k=0,…,4.

The Implicit Convex Feasibility Problem and Its Application to Adaptive Image Denoising

The implicit convex feasibility problem attempts to find a point in the intersection of a finite family of convex sets, some of which are not explicitly determined but may vary. We develop simultaneous and sequential projection methods capable of handling such problems and demonstrate their applicability to image denoising in a specific medical imaging situation. By allowing the variable sets to undergo scaling, shifting and rotation, this work generalizes previous results wherein the implicit convex feasibility problem was used for cooperative wireless sensor network positioning where sets are balls and their centers were implicit.

Colourful and Fractional (p,q)-theorems

Let p≥q≥d+1 be positive integers and let ${\mathcal{F}}$ be a finite family of convex sets in ${\mathbb{R}}^{d}$ . Assume that the elements of ${\mathcal{F}}$ are coloured with p colours. A p-element subset of ${\mathcal{F}}$ is heterochromatic if it contains exactly one element of each colour. The family ${\mathcal{F}}$ has the heterochromatic (p,q)-property if in every heterochromatic p-element subset there are at least q elements that have a point in common. We show that, under the heterochromatic (p,q)-condition, some colour class can be pierced by a finite set whose size we estimate from above in terms of d,p, and q. This is a colourful version of the famous (p,q)-theorem. (We prove a colourful variant of the fractional Helly theorem along the way.) A fractional version of the same problem is when the (p,q)-condition holds for all but an α fraction of the p-tuples in ${\mathcal{F}}$ . We show that, in the case that d=1, all but a β fraction of the elements of ${\mathcal{F}}$ can be pierced by p−q+1 points. Here β depends on α and p,q, and β→0 as α goes to zero.

An acceleration scheme for cyclic subgradient projections method

An algorithm for solving convex feasibility problem for a finite family of convex sets is considered. The acceleration scheme of De Pierro (em Methodos de projecao para a resolucao de sistemas gerais de equacoes algebricas lineares. Thesis (tese de Doutoramento), Instituto de Matematica da UFRJ, Cidade Universitaria, Rio de Janeiro, Brasil, 1981), which is designed for simultaneous algorithms, is used in the algorithm to speed up the fully sequential cyclic subgradient projections method. A convergence proof is presented. The advantage of using this strategy is demonstrated with some examples.

NEW RESULTS FOR T ( k )‐FAMILIES IN THE PLANE

A line that intersects every member of a finite family F of convex sets in the plane is called a common transversal to F . In this paper we study some basic properties of T ( k )-families: finite families of convex sets in the plane in which every subfamily of size at most k admits a common transversal. It is known that a T ( k )-family admits a partial transversal of size α ∣ F ∣ for some constant α ( k ) which is independent of F . Here it will be shown that (2/( k ( k −1))) 1/( k −2) ≤ α ( k )≤(( k −2)/( k −1)), which are the best bounds to date.

Piercing Families of Convex Sets with the d-Intersection Property in ℝ d

In this paper, we consider finite families of convex sets in ℝ d such that every d or fewer sets of the family have a common point. For some families of this type, we give upper bounds on the size of a finite set intersecting all sets of the family.

On the Chromatic Number of Intersection Graphs of Convex Sets in the Plane

Let $G$ be the intersection graph of a finite family of convex sets obtained by translations of a fixed convex set in the plane. We show that every such graph with clique number $k$ is $(3k-3)$-degenerate. This bound is sharp. As a consequence, we derive that $G$ is $(3k-2)$-colorable. We show also that the chromatic number of every intersection graph $H$ of a family of homothetic copies of a fixed convex set in the plane with clique number $k$ is at most $6k-6$.

A Helly Type Conjecture

A family of sets is Π n , or n -pierceable, if there exists a set of n points such that each member of the family contains at least one of them. It is Π k n if every subfamily of size k or less is Π n . Helly's theorem is one of the fundamental results in Combinatorial Geometry. It asserts, in the special case of finite families of convex sets in the plane, that Π 3 1 implies Π 1 . However, there is no k such that Π k 2 implies 2 -pierceability for all finite families of convex sets in the plane. It is therefore natural to propose the following:

Generalized variational inequalities and fixed point theorems

In this paper, from a well-known fixed point theorem of Idzik [l], we deduce new variational inequality for not-necessarily locally convex topological vector spaces and fiied point theorems for generalized upper hemicontinuous multimaps. Consequently, all of the results in Kim and Tan [2], Ding [3], and some others are substantially extended and improved. Moreover, our new fixed point theorems on generalized upper hemicontinuous multimaps include a large number of historically well-known extensions of the Brouwer or Kakutani theorems. Throughout this paper, a t.v.s is a Hausdorff topological vector space, and co denotes the convex hull. A multimap or map T: X + Y is a function from X into the power set of Y with nonempty values, and x E T-y if and only if y E TX. For topological spaces X and Y, a map T: X d Y is said to be closed if its graph Gr( T) = [(x, y) : x E X, y E TX) is closed in X x Y, and compact if the closure 7(X) of its range T(x) is compact in Y. A subset B of a t.v.s. E is convexly totally bounded (c.t.b) if for every neighborhood V of the origin 0 of E, there exist a finite subset (Xi : i E I) of E and a finite family of convex sets (Ci : i E I) such that Ci C V for each i E Z and B C U (Xi + Ci : i E I). See [ 11. Let V be the fundamental system of neighborhoods of the origin 0 in E. We recall that a set K c E is locally convex if for every x E K and every V E 9, there exists U E V such that co((x + U) n K) c x + K We say that K c E is of 2 type if for every I/ E V there exists U E V such that co(U n (K - ZQ) C K See [4]. The following are noted by Idzik [l]: 1. Every compact set in a locally convex t.v.s is c.t.b. 2. In a locally convex t.v.s., every subset is of Z type and is locally convex. 3. If K is a compact subset of a t.v.s. such that either K is locally convex or of Z type, then it is c.t.b. 4. Every compact convex subset of a t.v.s. Eon which E* separates points is c.t.b. See 151. The following is a particular form of Idzik [l , Theorem 4.31.

A parallel subgradient projections method for the convex feasibility problem

An iterative method to solve the convex feasibility problem for a finite family of convex sets is presented. The algorithm consists in the application of a generalization of an acceleration procedure introduced by De Pierro, in a parallel version of the Subgradient Projections Method proposed by Censor and Lent. The generated sequence is shown to converge for any starting point. Some numerical results are presented.

Characterization of f-vectors of families of convex sets in [formula omitted]d part II: Sufficiency of Eckhoff's conditions

Let K = { K 1,…, K n } be a family of n convex sets in R d . For 0⩽ i< n denote by f i the number of subfamilies of K of size i + 1 with non-empty intersection. The vector f( K) = ( f 0, f 1,…) is called the f-vector of K. In 1973, Eckhoff proposed a characterization of the set of f-vectors of finite families of convex sets in R d by a system of inequalities. In part I we proved the necessity of Eckhoffs inequalities and here we prove their sufficiency.

An interior points algorithm for the convex feasibility problem

The problem of finding a point in the intersection of a finite family of convex sets in the Euclidean space R″ is considered here. We present a general algorithmic scheme which employs projections onto separating hyperplanes instead of projections onto the convex sets. This scheme includes the method of successive projections of Gubin et al., USSR Comp. Math. and Math. Phys. 7 (1967), 1–24, as a special case. A different realization proposed here is capable of handling the problem when the sets are solid and an interior point of each set is available. This alternative algorithm may, in certain cases, be more attractive than the method of Gubin et al.

Recent results on separation of convex sets1

In this paper a survey about the following directions of generalizations for separation theorems involving some new results are given: Separation of finite families of convex sets; separation of sets in product spaces; separation of convex sets in projective spaces; separation of convex sets in convexity spaces. As a basis for all these considerations vector spaces without topology are used.