Category Archives: Convexity

Buffon’s Needle and the Perimeter of Planar Sets of Constant Width

Here is an answer to “Test your intuition (8)”. (Essentially the answer posed by David Eppstein.) (From Wolfram Mathworld) Buffon’s needle problem asks to find the probability that a needle of length will land on a line, given a floor … Continue reading

Posted in Convexity, Probability, Test your intuition | Tagged | 5 Comments

Test Your Intuition (8)

Consider all planar sets  A with constant width 1. Namely, in every direction, the distance between the two parallel lines that touch A from both sides is 1. We already know that there exists such sets other than the circle … Continue reading

Posted in Convexity, Test your intuition | Tagged | 14 Comments

Raigorodskii’s Theorem: Follow Up on Subsets of the Sphere without a Pair of Orthogonal Vectors

Andrei Raigorodskii (This post follows an email by Aicke Hinrichs.) In a previous post we discussed the following problem: Problem: Let be a measurable subset of the -dimensional sphere . Suppose that does not contain two orthogonal vectors. How large … Continue reading

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A Little Story Regarding Borsuk’s Conjecture

Jeff Kahn Jeff and I worked on the problem for several years. Once he visited me with his family for two weeks. Before the visit I emailed him and asked: What should we work on in your visit? Jeff asnwered: … Continue reading

Posted in Convexity, Taxi-and-other-stories | Tagged , | 7 Comments

Test Your Intuition (5)

  (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?

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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

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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 , , , | 7 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

Posted in Combinatorics, Convexity, Open problems | Tagged , , | 21 Comments

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

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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 in … Continue reading

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