A Sufficient Condition for the Existence of Large Empty Convex Polygons

Abstract. Let P be a set of points in general position in the plane. We say that P is k -convex if no triangle determined by P contains more than k points of P in the interior. We say that a subset A of P in convex position forms an empty polygon (in P ) if no point of P \ A lies in the convex hull of A . We show that for any k,n there is an N=N(k,n) such that any k -convex set of at least N points in general position in the plane contains an empty n -gon. We also prove an analogous statement in R ^d for each odd d≥ 3 .     

A note on the upper bound for disjoint convex partitions

Let n(k, l,m), k ≤ l ≤ m, be the smallest integer such that any finite planar point set which has at least n(k, l,m) points in general position, contains an empty convex k-hole, an empty convex l-hole and an empty convex m-hole, in which the three holes are pairwise disjoint. In this article, we prove that n(4, 4, 5) ≤ 16.     

Large empty convex polygons in k-convex sets

A point set is k-convex if there are at most k points of in any triangle having its vertices in . Károlyi, Pach and Tóth [6] showed that if a 1-convex set has sufficiently many points, then it contains an arbitrarily large emtpy convex polygon. They also constructed exponentially large 1-convex sets that contain no empty convex n-gons. Here we shall give an exponential upper bound to the number of points needed. Valtr [8] proved a similar result for k-convex sets. In this paper we improve his upper bound and give an elementary proof of the statement.     

Finding Sets of Points without Empty Convex 6-Gons

   Abstract. Erd?s asked whether every large enough set of points in general position in the plane contains six points that form a convex 6-gon without any points from the set in its interior. In this note we show how a set of 29 points was found that contains no empty convex 6-gon. To this end a fast incremental algorithm for finding such 6-gons was designed and implemented and a heuristic search approach was used to find promising sets. The experiments also led to two observations that might be useful in proving that large sets always contain an empty convex 6-gon.     

On empty convex polygons in a planar point set

Let P be a set of n points in general position in the plane. Let X_k(P) denote the number of empty convex k-gons determined by P. We derive, using elementary proof techniques, several equalities and inequalities involving the quantities X_k(P) and several related quantities. Most of these equalities and inequalities are new, except for a few that have been proved earlier using a considerably more complex machinery from matroid and polytope theory, and algebraic topology. Some of these relationships are also extended to higher dimensions. We present several implications of these relationships, and discuss their connection with several long-standing open problems, the most notorious of which is the existence of an empty convex hexagon in any point set with sufficiently many points.     

Partitioning a graph into convex sets

Let G be a finite simple graph. Let S⊆V(G), its closed interval I[S] is the set of all vertices lying on shortest paths between any pair of vertices of S. The set S is convex if I[S]=S. In this work we define the concept of a convex partition of graphs. If there exists a partition of V(G) into p convex sets we say that G is p-convex. We prove that it is NP-complete to decide whether a graph G is p-convex for a fixed integer p≥2. We show that every connected chordal graph is p-convex, for 1≤p≤n. We also establish conditions on n and k to decide if the k-th power of a cycle C_n is p-convex. Finally, we develop a linear-time algorithm to decide if a cograph is p-convex.     

Constructions from empty polygons

Let P denote a finite set of points, in general position in the plane. In this note we study conditions which guarantee that P contains the vertex set of a convex polygon that has exactly k points of P in its interior.     

On an empty triangle with the maximum area in planar point sets

Let P be a finite set of points in general position in the plane. We evaluate the ratio between the maximum area of an empty triangle of P and the area of the convex hull of P.     

The Empty Hexagon Theorem

Let P be a finite set of points in general position in the plane. Let C(P) be the convex hull of P and let Cⁱ_P be the ith convex layer of P. A minimal convex set S of P is a convex subset of P such that every convex set of P ∩ C(S) different from S has cardinality strictly less than |S|. Our main theorem states that P contains an empty convex hexagon if C¹_P is minimal and C⁴_P is not empty. Combined with the Erdos-Szekeres theorem, this result implies that every set P with sufficiently many points contains an empty convex hexagon, giving an affirmative answer to a question posed by Erdos in 1977.     

The k Most Frequent Distances in the Plane

   Abstract. A new upper bound is shown for the number of incidences between n points and n families of concentric circles in the plane. As a consequence, it is shown that the number of the k most frequent distances among n points in the plane is f _n (k)=O(n ^1.4571 k ^.6286 ) improving on an earlier bound of Akutsu, Tamaki, and Tokuyama.     

Finding Sets of Points without Empty Convex 6-Gons

Abstract. Erd?s asked whether every large enough set of points in general position in the plane contains six points that form a convex 6-gon without any points from the set in its interior. In this note we show how a set of 29 points was found that contains no empty convex 6-gon. To this end a fast incremental algorithm for finding such 6-gons was designed and implemented and a heuristic search approach was used to find promising sets. The experiments also led to two observations that might be useful in proving that large sets always contain an empty convex 6-gon.     

Finding Convex Sets in Convex Position

Let F{\cal{F}} denote a family of pairwise disjoint convex sets in the plane. F{\cal{F}} is said to be in convex position, if none of its members is contained in the convex hull of the union of the others. For any fixed k 3 5k\ge5, we give a linear upper bound on P_k(n)P_k(n), the maximum size of a family F{\cal{F}} with the property that any k members of F{\cal{F}} are in convex position, but no n are.     

On directional convexity

Motivated by problems from calculus of variations and partial differential equations, we investigate geometric properties of D-convexity. A function f: R ^d → R is called D-convex, where D is a set of vectors in R ^d, if its restriction to each line parallel to a nonzero v ∈ D is convex. The D-convex hull of a compact set A ⊂ R ^d, denoted by co^D(A), is the intersection of the zero sets of all nonnegative D-convex functions that are zero on A. It also equals the zero set of the D-convex envelope of the distance function of A. We give an example of an n-point set A ⊂ R ² where the D-convex envelope of the distance function is exponentially close to zero at points lying relatively far from co ^D(A), showing that the definition of the D-convex hull can be very nonrobust. For separate convexity in R ³ (where D is the orthonormal basis of R ³), we construct arbitrarily large finite sets A with co ^D(A) ≠ A whose proper subsets are all equal to their D-convex hull. This implies the existence of analogous sets for rank-one convexity and for quasiconvexity on 3 × 3 (or larger) matrices. This research was supported by Charles University Grants No. 158/99 and 159/99.     

Aperture-Angle and Hausdorff-Approximation of Convex Figures

The aperture angle α(x,Q) of a point x ∉ Q in the plane with respect to a convex polygon Q is the angle of the smallest cone with apex x that contains Q. The aperture angle approximation error of a compact convex set C in the plane with respect to an inscribed convex polygon Q⊂C is the minimum aperture angle of any x∈C∖Q with respect to Q. We show that for any compact convex set C in the plane and any k>2, there is an inscribed convex k-gon Q⊂C with aperture angle approximation error . This bound is optimal, and settles a conjecture by Fekete from the early 1990s. The same proof technique can be used to prove a conjecture by Brass: If a polygon P admits no approximation by a sub-k-gon (the convex hull of k vertices of P) with Hausdorff distance σ, but all subpolygons of P (the convex hull of some vertices of P) admit such an approximation, then P is a (k+1)-gon. This implies the following result: For any k>2 and any convex polygon P of perimeter at most 1 there is a sub-k-gon Q of P such that the Hausdorff-distance of P and Q is at most  . This research was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-311-D00763). NICTA is funded through the Australian Government’s Backing Australia’s Ability initiative, in part through the Australian Research Council.     

An operator-theoretic approach to theorems of the Pick-Nevanlinna and Carathéodory types

In this paper we deal with singularities of the linear systems of plane curves passing through S, where S is a zerodimensional closed subscheme of degree n of P ²=P _k ² ,k an algebraically closed field of any characteristic. We determine the least degree of a nonsingular curve passing through S, when S is in uniform position. This paper was written while the author was member of C.N.R., Sez. 3 of G.N.S.A.G.A. and was supported by M.P.I. funds     

Empty Convex Hexagons in Planar Point Sets

Erdős asked whether every sufficiently large set of points in general position in the plane contains six points that form a convex hexagon without any points from the set in its interior. Such a configuration is called an empty convex hexagon. In this paper, we answer the question in the affirmative. We show that every set that contains the vertex set of a convex 9-gon also contains an empty convex hexagon.     

A Generalization of the Erdos - Szekeres Theorem to Disjoint Convex Sets

Let F denote a family of pairwise disjoint convex sets in the plane. F is said to be in convex position if none of its members is contained in the convex hull of the union of the others. For any fixed k≥ 3 , we estimate P _k (n) , the maximum size of a family F with the property that any k members of F are in convex position, but no n are. In particular, for k=3 , we improve the triply exponential upper bound of T. Bisztriczky and G. Fejes Tóth by showing that P ₃ (n) < 16 ⁿ . <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p437.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received March 27, 1997, and in revised form July 10, 1997.     

Generalizing Ham Sandwich Cuts to Equitable Subdivisions

We prove a generalization of the famous Ham Sandwich Theorem for the plane. Given gn red points and gm blue points in the plane in general position, there exists an equitable subdivision of the plane into g disjoint convex polygons, each of which contains n red points and m blue points. For g=2 this problem is equivalent to the Ham Sandwich Theorem in the plane. We also present an efficient algorithm for constructing an equitable subdivision. Received February 19, 1999, and in revised form June 3, 1999. {\it Online publication August\/} 18, 2000.     

The k Most Frequent Distances in the Plane

Abstract. A new upper bound is shown for the number of incidences between n points and n families of concentric circles in the plane. As a consequence, it is shown that the number of the k most frequent distances among n points in the plane is f _n (k)=O(n ^1.4571 k ^.6286 ) improving on an earlier bound of Akutsu, Tamaki, and Tokuyama.     

Restricted Point Configurations with Many Collinear {k}-Tuplets

   Abstract. Given k≥ 3 , denote by t' _k (N) the largest integer for which there is a set of N points in the plane, no k+1 of them on a line such that there are t' _k (N) lines, each containing exactly k of the points. Erdos (1962) raised the problem of estimating the order of magnitude of t' _k (N) . We prove that