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Category Archives: Convexity
Colorful Caratheodory Revisited
Janos Pach wrote me: ”I saw that you several times returned to the colored Caratheodory and Helly theorems and related stuff, so I thought that you may be interested in the enclosed paper by Holmsen, Tverberg and me, in … Continue reading
Lovasz’s Two Families Theorem
Laci and Kati This is the first of a few posts which are spin-offs of the extremal combinatorics series, especially of part III. Here we talk about Lovasz’s geometric two families theorem. 1. Lovasz’s two families theorem Here … Continue reading
Posted in Combinatorics, Convexity, Open problems
Tagged exterior algebras, Extremal combinatorics, shellability
4 Comments
Seven Problems Around Tverberg’s Theorem
Imre Barany, Rade Zivaljevic, Helge Tverberg, and Sinisa Vrecica Recall the beautiful theorem of Tverberg: (We devoted two posts (I, II) to its background and proof.) Tverberg Theorem (1965): Let be points in , . Then there is a partition of … Continue reading
Test Your Intuition (2)
Question: Let be the cube in centered at the origin and having -dimensional volume equal to one. What is the maximum -dimensional volume of when is a hyperplane? Can you guess the behavior of when ? Can you guess the plane which … Continue reading
Sarkaria’s Proof of Tverberg’s Theorem 2
Karanbir Sarkaria 4. Sarkaria’s proof: Tverberg’s theorem (1965): Let be points in , . Then there is a partition of such that . Proof: We can assume that . First suppose that the points belong to the -dimensional affine space … Continue reading
Sarkaria’s Proof of Tverberg’s Theorem 1
Helge Tverberg Ladies and gentlemen, this is an excellent time to tell you about the beautiful theorem of Tverberg and the startling proof of Sarkaria to Tverberg’s theorem (two parts). A good place to start is Radon’s theorem. 1. The theorems of … Continue reading
Helly, Cayley, Hypertrees, and Weighted Enumeration III
This is the third and last part of the journey from a Helly type conjecture of Katchalski and Perles to a Cayley’s type formula for “hypertrees”. (On second thought I decided to divide it into two devoting the second to probabilistic questions.) … Continue reading
Posted in Combinatorics, Convexity, Open problems, Probability
5 Comments
Helly’s Theorem, “Hypertrees”, and Strange Enumeration II: The Formula
In the first part of this post we discussed an appealing conjecture regaring an extension of Cayley’s counting trees formula. The number of d-dimensional “hypertrees” should somehow add up to . But it was not clear to us which complexes we want … Continue reading
Posted in Combinatorics, Convexity
Tagged Cayley theorem, Helly type theorems, Topological combinatorics
2 Comments
Helly’s Theorem, “Hypertrees”, and Strange Enumeration I
1. Helly’s theorem and Cayley’s formula Helly’s theorem asserts: For a family of n convex sets in , n > d, if every d+1 sets in the family have a point in common then all members in the family have a point in common. … Continue reading
Posted in Combinatorics, Convexity
Tagged Cayley theorem, Helly Theorem, Simplicial complexes, Topological combinatorics, Trees
5 Comments
Five Open Problems Regarding Convex Polytopes
The problems 1. The conjecture A centrally symmetric d-polytope has at least non empty faces. 2. The cube-simplex conjecture For every k there is f(k) so that every d-polytope with has a k-dimensional face which is either a simplex … Continue reading
