Imre Bárány

A generalization of carathéodory's theorem

The following theorem is proved. If the sets V1,..., Vn+1@?R^n and a @e@?^n^+^1i=1 conv Vi, then there exist elements vi@eVi (i=1...,n+1) such that a@econv{v1,...,vn+1}. This is generalization of Caratheodorys theorem. By applying this and similar results some open questions are answered.

On the number of halving planes

Let <italic>S</italic> ⊂ R<supscrpt>3</supscrpt> be an <italic>n</italic>-set in general position. A plane containing three of the points is called a halving plane if it dissects <italic>S</italic> into two parts of equal cardinality. It is proved that the number of halving planes is at most <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>2.998</supscrpt>).nAs a main tool, for every set <italic>Y</italic> of <italic>n</italic> points in the plane a set <italic>N</italic> of size <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>4</supscrpt>) is constructed such that the points of <italic>N</italic> are distributed almost evenly in the triangles determined by <italic>Y</italic>.

ON THE NUMBER OF HALVING PLANES

LetS ⊂ℝ3 be ann-set in general position. A plane containing three of the points is called a halving plane if it dissectsS into two parts of equal cardinality. It is proved that the number of halving planes is at mostO(n2.998).As a main tool, for every setY ofn points in the plane a setN of sizeO(n4) is constructed such that the points ofN are distributed almost evenly in the triangles determined byY.

Point Selections and Weak ε-Nets for Convex Hulls

One of our results: Let X be a finite set on the plane, 0 < e < 1. Then there exists a set F (a weak e-net) of size at most 7/e such that every convex set containing at least e|X| elements of X intersects F . Note that the size of F is independent of the size of X. 1Research supported by a United Stat es Isreal BSF grant 2On leave from the Mathematica l Institute of the Hungarian Academy of Sciences, POB 127, 1364 Budapest, Hungar y. Supported by the Program in Discrete Mathematics and its Applications at Yale , NSF grant 8901484, and Hungarian National Science Foundation grant No. 1812. 3This research was done whil e the author visited the Department of Mathematics at Massachusetts Institute of Technology. Supported in part by Hungarian National Science Foundation grant No . 1812. 4Supported in part by NSF g rants DMS–86–06225 and AFOSR–0271. Submitted to Combinatori cs, Probability, and Computation October 1991 1 2 POINT SELECTIONS AND WEAK e-NETS FOR CONVEX HULLS

Random polytopes in a convex polytope, independence of shape, and concentration of vertices

Write ~.-d for the set of all convex bodies (convex compact sets with nonempty interior) in ~d. Define o@g~l d as the set of those K E 5 b ~d with vol K = 1. Fix K E .~g-i d and choose points X l , . . . , x~ E K randomly, independently, and according to the uniform distribution on K. Then K,~ = c o n v ( x l , . . . , xn} is a random polytope in K . Write E(K, n) for the expectation of the random variable v o l ( K K n ) . E(K, n) shows how well K,~ approximates K in volume on the average. Groemer [Grl] proved that, among all convex bodies K E o@g~l d, the ellipsoids are approximated worst, i.e.

Colourful Linear Programming and its Relatives

We consider the following Colourful generalization of Linear Programming: given sets of points S1,..., Sk ⊂ Rd, referred to as colours, and a point b ∈ Rd, decide whether there is a colourfulT = {s1,..., sk} such that b ∈ convT, and if there is one, find it. Linear Programming is obtained by taking k = d + 1 and S1 =... = Sd+1. If k = d + 1 and b ∈ ∩i=1d+1 convSi then a solution always exists: we describe an efficient iterative approximation algorithm for this problem, that finds a colourful T whose convex hull contains a point e-close to b, and analyze its real arithmetic and Turing time complexities. In contrast, we show that Colourful Linear Programming is strongly NP-complete. We consider a class of linear algebraic relatives of Colourful Linear Programming, and give a computational complexity classification of the related decision and counting problems that arise. We also introduce and discuss the complexity of a hierarchy of w1, w2-Matroid-Basis-Nonbasis problems, and give an application of Colourful Linear Programming to the algorithmic problems of Tverbergs theorem in combinatorial geometry.

On the convex hull of the integer points in a disc

Let P_{r} denote the convex hull of the integer points in the disc of radius r. We prove that the number of vertices of P_{r} is essentially r^{2/3} as r approaches infinity.

On Integer Points in Polyhedra: A Lower Bound

Given a polyhedronP⊂ℝ we writePI for the convex hull of the integral points inP. It is known thatPI can have at most135-2 vertices ifP is a rational polyhedron with size φ. Here we give an example showing thatPI can have as many as Ω(ϕn−1) vertices. The construction uses the Dirichlet unit theorem.

On the number of convex lattice polygons

Note: Professor Pachs number: [093] Reference DCG-ARTICLE-2008-017doi:10.1017/S0963548300000341 Record created on 2008-11-17, modified on 2017-05-12

The complex of maximal lattice free simplices

The simplicial complexK(A) is defined to be the collection of simplices, and their proper subsimplices, representing maximal lattice free bodies of the form (x: Ax⩽b), withA a fixed generic (n + 1) ×n matrix. The topological space associated withK(A) is shown to be homeomorphic to ℝn, and the space obtained by identifying lattice translates of these simplices is homeorphic to then-torus.