# 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.

Posted in Convexity, Open problems | Tagged | 2 Comments

## 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

Posted in Combinatorics, Convexity | Tagged | 7 Comments

## 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 | 3 Comments

## 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 | | 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

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