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Category Archives: Convexity
Around Borsuk’s Conjecture 3: How to Save Borsuk’s conjecture
Borsuk asked in 1933 if every bounded set K of diameter 1 in can be covered by d+1 sets of smaller diameter. A positive answer was referred to as the “Borsuk Conjecture,” and it was disproved by Jeff Kahn and me in 1993. … Continue reading
A Weak Form of Borsuk Conjecture
Problem: Let P be a polytope in with n facets. Is it always true that P can be covered by n sets of smaller diameter? I also asked this question over mathoverflow, with some background and motivation.
Around Borsuk’s Conjecture 1: Some Problems
Greetings to all! Karol Borsuk conjectured in 1933 that every bounded set in can be covered by sets of smaller diameter. In a previous post I described the counterexample found by Jeff Kahn and me. I will devote a few posts … Continue reading
The Combinatorics of Cocycles and Borsuk’s Problem.
Cocycles Definition: A cocycle is a collection of subsets such that every set contains an even number of sets in the collection. Alternative definition: Start with a collection of sets and consider all sets that contain an odd number of members … Continue reading
Nerves of Convex Sets – A Recent Result by Martin Tancer
Martin Tancer recently found a very beautiful proof that finite projective planes can’t be represented by convex sets in any fixed dimension. This was asked in the paper entitled “Transversal numbers for hypergraphs arising in geometry” by Noga Alon, Gil … Continue reading
Posted in Convexity
2 Comments
Optimal Colorful Tverberg’s Theorem by Blagojecic, Matschke, and Ziegler
Pavle Blagojevic, Benjamin Matschke, and Guenter Ziegler settled the “colorful Tverberg’s conjecture.” (Problem 6 in this post.) This gives a sharp version for Zivaljevic and Vrecica theorem, and crossed the “connectivity of chessboard complexes barrier”. Here is the link to the breakthrough paper.
Posted in Convexity
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Igor Pak’s “Lectures on Discrete and Polyhedral Geometry”
Here is a link to Igor Pak’s book on Discrete and Polyhedral Geometry (free download) . And here is just the table of contents. It is a wonderful book, full of gems, contains original look on many important directions, things that … Continue reading
Posted in Book review, Convex polytopes, Convexity
Tagged Convex polytopes, Convexity, Igor Pak, rigidity
4 Comments
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
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
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