On a convex operator for finite sets

One classical result of Freimann gives the optimal lower bound for the cardinality of A + A if A is a d-dimensional finite set in R d . Matolcsi and Ruzsa have recently generalized this lower bound to |A + kB| if B is d-dimensional, and A is contained in the convex hull of B. We characterize the equality case of the Matolcsi-Ruzsa bound. The argument is based partially on understanding triangulations of polytopes.

2020

We study R^2⊕R-separately convex hulls of finite sets of points in R^3, as introduced in <cit.>. When R^3 is considered as a certain subset of 3× 2 matrices, this notion of convexity corresponds to rank-one convex convexity K^rc. If R^3 is identified instead with a subset of 2× 3 matrices, it actually agrees with the quasiconvex hull, due to a recent result <cit.>. We introduce "2+1 complexes", which generalize T_n constructions. For a finite set K, a "2+1K-complex" is a 2+1 complex whose extremal points belong to K. The "2+1-complex convex hull of K", K^cc, is the union of all 2+1K-complexes. We prove that K^cc is contained in the 2+1 convex hull K^rc. We also consider outer approximations to 2+1 convexity based in the locality theorem <cit.>. Starting with a crude outer approximation we iteratively chop off "D-prisms". For the examples in <cit.>, and many others, this procedure reaches a "2+1K-complex" in a ...

1997

A known result in combinatorial geometry states that any collection P n of points on the plane contains two such that any circle containing them contains n/c elements of P n , c a constant. We prove: Let be a family of n noncrossing compact convex sets on the plane, and let S be a strictly convex compact set. Then there are two elements S i , S j of such that any set S homothetic to S that contains them contains n/c elements of , c a constant (S is homothetic to S if S = λS + v, where λ is a real number greater than 0 and v is a vector of 2). Our proof method is based on a new type of Voronoi diagram, called the "closest covered set diagram" based on a convex distance function. We also prove that our result does not generalize to higher dimensions; we construct a set of n disjoint convex sets in 3 such that for any nonempty subset H of there is a sphere S H containing all the elements of H , and no other element of .

Acta Scientiarum Mathematicarum, 2017

Very recently Richter and Rogers proved that any convex geometry can be represented by a family of convex polygons in the plane. We shall generalize their construction and obtain a wide variety of convex shapes for representing convex geometries. We present an Erdős-Szekeres type obstruction, which answers a question of Czédli negatively, that is general convex geometries cannot be represented with ellipses in the plane. Moreover, we shall prove that one cannot even bound the number of common supporting lines of the pairs of the representing convex sets. In higher dimensions we prove that all convex geometries can be represented with ellipsoids.

2001

Let R and B be point sets such that R ∪ B is in general position. We say that B is enclosed by R if there is a simple polygon P with vertex set R such that all the elements in B belong to the interior of P. In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv (R)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56|Conv (B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p 1 ,. .. , p n and S a set of m points contained in the interior of P , m ≤ n − 1. Then there is a convex decomposition {P 1 ,. .. , P n } of P such that all points from S lie on the boundaries of P 1 ,. .. , P n , and each P i contains a whole edge of P on its boundary.

Graphs and Combinatorics, 2009

Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P. In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv (B)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56|Conv (B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p 1 ,. .. , p n and S a set of m points contained in the interior of P , m ≤ n − 1. Then there is a convex decomposition {P 1 ,. .. , P n } of P such that all points from S lie on the boundaries of P 1 ,. .. , P n , and each P i contains a whole edge of P on its boundary.

Journal of Combinatorial Theory, Series B, 1971

Computational Geometry, 2013

Given a set Σ of spheres in E d , with d ≥ 3 and d odd, having a constant number of m distinct radii ρ 1 , ρ 2 , . . . , ρ m , we show that the worst-case combinatorial complexity of the convex hull of Σ is Θ( 1≤i =j≤m n i n ⌊ d 2 ⌋ j ), where n i is the number of spheres in Σ with radius ρ i . To prove the lower bound, we construct a set of Θ(n 1 +n 2 ) spheres in E d , with d ≥ 3 odd, where n i spheres have radius ρ i , i = 1, 2, and ρ 2 = ρ 1 , such that their convex hull has combinatorial complexity Ω(n 1 n

Proceedings of the 27th annual ACM symposium on Computational geometry - SoCG '11, 2011

Given a set Σ of spheres in E d , with d ≥ 3 and d odd, having a fixed number of m distinct radii ρ 1 , ρ 2 , . . . , ρ m , we show that the worst-case combinatorial complexity of the convex hull

Journal of Physics a Mathematical and Theoretical, 2009

Column-convex polygons were first counted by area several decades ago, and the result was found to be a simple, rational, generating function. In this work we generalize that result. Let a p-column polyomino be a polyomino whose columns can have 1, 2, ..., p connected components. Then column-convex polygons are equivalent to 1-convex polyominoes. The area generating function of even the simplest generalization, namely 2-column polyominoes, is unlikely to be solvable. We therefore define two classes of polyominoes which interpolate between column-convex polygons and 2-column polyominoes. We derive the area generating functions of those two classes, using extensions of existing algorithms. The growth constants of both classes are greater than the growth constant of column-convex polyominoes. Rather tight lower bounds on the growth constants complement a comprehensive asymptotic analysis.