Several generalizations of Tverberg’s theorem

  title={Several generalizations of Tverberg’s theorem},
  author={John R. Reay},
  journal={Israel Journal of Mathematics},
  • J. R. Reay
  • Published 1 September 1979
  • Mathematics
  • Israel Journal of Mathematics
AbstractIn a generalization of Radon’s theorem, Tverberg showed that each setS of at least (d+1) (r − 1)+1 points inRdhas anr-partition into (pair wise disjoint) subsetsS =S1 ∪ … ∪Srso that $$\bigcap\nolimits_i^r {\underline{\underline {}} } _1 $$ convSi# Ø. This note considers the following more general problems: (1) How large mustS σRdbe to assure thatS has anr-partitionS=S1∪ … ∪Srso that eachn members of the family {convSi∼i-1r have non-empty intersection, where 1<=n<=r. (2) How large mustS… 
Tverberg-Type Theorems with Altered Intersection Patterns (Nerves)
The main results of this paper demonstrate that Tverberg's theorem is just a special case of a more general situation, where other simplicial complexes must always arise as nerve complexes, as soon as the number of points is large enough.
Some variations on Tverberg’s theorem
Define T(d, r) = (d + 1)(r - 1) + 1. A well known theorem of Tverberg states that if n ≥ T(d, r), then one can partition any set of n points in Rd into r pairwise disjoint subsets whose convex hulls
Tverberg Partitions of Points on the Moment Curve
The set A is “usually” in “strong general position” (SGP) and some relationships among three notions are investigated: strong general position, Tverberg’s theorem and the moment curve.
A generalization of Tverberg's Theorem
The well know theorem of Tverberg states that if n > (d+1)(r-1) then one can partition any set of n points in R^d to r disjoint subsets whose convex hulls have a common point. The numbers T(d,r) = (d
Helly, Radon, and Carathéodory Type Theorems
This chapter discusses applications and generalizations of the classical theorems of Helly, Radon, and Caratheodory, as well as their ramifications in the context of combinatorial convexity theory.
On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture
It is proved that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections, a higher-dimensional analogue of Conway's thrackle conjecture.
On Eckhoff's conjecture for Radon numbers; or how far the proof is still away
Eckhoff's conjecture for the Τ-Radon numbers r(Τ) of a convexity space. (X,C) says r(Τ) ≦ (r−1)(Τ−1)+1, with r = r(2). The main result of this note is that Eckhoff's conjecture is true in case ¦X¦ ≦
Robust Tverberg and Colourful Carathéodory Results via Random Choice
  • Pablo Soberón
  • Mathematics, Computer Science
    Combinatorics, Probability and Computing
  • 2017
Borders are given for the smallest integer N = N(t,d,r) such that for any N points in ℝd, there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect.
A Tverberg-type generalization of the Helly number of a convexity space
In 1966 H. Tverberg gave a far reaching generalization of the well-known classical theorem of J. Radon. In this paper a similar generalization of the classical Helly theorem is given and it is shown
The m-core properly contains the m-divisible points in space
  • D. Avis
  • Mathematics, Computer Science
    Pattern Recognit. Lett.
  • 1993
A counterexample for n =9, d =3, m =3 is given to prove the conjecture that conv D m ( S)= C m (S) is known to be true for d =2.


A Generalization of Radon's Theorem
and proved it in the case n = 2. In the case n > 2, Birch proved a weaker result, with r(n+l) — n replaced by rn(n+l) — n — n+ 1. This was, for most r and n, an improvement of the earlier result, by
On a class of convex polytopes
Let ℘ denote the class of convex polytopesP having the following property: IfQ1 andQ2 are any subpolytopes ofP with no vertex in common, thenQ1 ∩Q2 is either empty or a single point. A
On 3 N points in a plane
In this note I prove the following theorem:Theorem 1. Given 3N points in a plane, we can divide them into N triads such that, when we form a triangle with the points of each triad, the N triangles