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Category Archives: Convexity
(Not such a set) consider a planar set A with the following property. In every direction, the distance between the two parallel lines that touch A from both sides is the same! Must A be a circle?
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
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
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
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
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
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 Radon, … Continue reading
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
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
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