Borsuk conjecture

The Borsuk conjecture is a mathematical conjecture from the field of geometry . It is about the question of how many parts a given quantity of limited diameter has to be broken down into so that each part has a really smaller diameter. The question posed by Karol Borsuk in 1933 and later referred to as a conjecture, whether you can always get by with parts in dimensions , was answered negatively 60 years later. ${\displaystyle n}$ ${\displaystyle n+1}$

A segment , a triangle and a tetrahedron can be broken down into 2, 3 or 4 smaller parts.

The presumption

In n-dimensional space , the Euclidean norm can be used to define the diameter of a set as (maximum distance between two points of the set). ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle E\subset \mathbb {R} ^{n}}$ ${\displaystyle \sup\{\|x-y\|;\,x,y\in E\}}$

One can now try to divide the quantity into subsets so that each part has a really smaller diameter than . The question arises as to how many subsets are required for this. ${\displaystyle E}$ ${\displaystyle E=E_{1}\cup \ldots \cup E_{k}}$ ${\displaystyle E_{i}}$ ${\displaystyle E}$ ${\displaystyle E_{i}}$

As the regular, n-dimensional simplex shows, in general at least quantities are required, because the corners all have the same distance, which is equal to the diameter. A subset of really smaller diameter can therefore contain at most one corner, that is, one needs at least as many subsets as there are corners, and one has of them . As the adjacent drawing for dimensions 1, 2 and 3 clearly shows, with the Simplex you can actually manage with subsets. Karol Borsuk completed his thesis " Three Sentences about the n-dimensional sphere ", in which he dealt with the decomposition of spheres, as follows: ${\displaystyle n+1}$ ${\displaystyle n+1}$ ${\displaystyle n+1}$ ${\displaystyle n+1}$

The following question remains open: Can every bounded subset E of space be decomposed into (n + 1) sets, each of which has a diameter smaller than E? ${\displaystyle R^{n}}$

The presumption that this question should be answered in the affirmative came to be known as the Borsuk conjecture and remained open for 60 years.

refutation

The assumption was confirmed in the room in 1955. It may therefore come as a surprise that the Borsuk Conjecture turns out to be false in higher dimensions. In 1993 Jeff Kahn and G. Kalai showed that at least subsets are required for sufficiently large dimensions , which refuted the Borsuk conjecture, because it grows faster than . A concrete counterexample was found by A. Nilli in the 964-dimensional space, later another by A. Hinrichs and C. Richter in the 298-dimensional space. Today it is known that the Borsuk conjecture is wrong for dimensions from 64 onwards. The question of the smallest dimension from which the Borsuk conjecture no longer applies is open. ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle n}$ ${\displaystyle (1,2)^{\sqrt {n}}}$ ${\displaystyle (1,2)^{\sqrt {n}}}$ ${\displaystyle n+1}$

Individual evidence

1. ↑ K. Borsuk: Three sentences about the n-dimensional sphere (PDF; 1.1 MB) , Fundamenta Mathematica (1933), Volume 20, pages 177-190
2. ^ HG Eggleston: Covering a three-dimensional set with sets of smaller diameter , J. London Math. Society (1955), Volume 30, pages 11-24
3. ↑ Kahn, Kalai A Counterexample to Borsuks conjecture , Bulletin American Mathematical Society, Vol. 29, 1993, pp. 60-62
4. ^ A. Nilli: On Borsuk's problem , Jerusalem Combinatorics '93, Contemporary Mathematics 178, AMS 1994, pp. 209-210
5. ^ A. Hinrichs and C. Richter: New sets with large Borsuk numbers , Discrete Math. (2003), Volume 270, Pages 137-147
6. ^ Andriy V. Bondarenko: On Borsuk's conjecture for two-distance sets
7. ↑ Thomas Jenrich: A 64-dimensional two-distance counterexample to Borsuk's conjecture

swell

This topic is presented in

• M. Aigner, GM Ziegler: The Book of Evidence Springer, Berlin 2002, ISBN 3-540-42535-7 (3rd edition: ISBN 978-3-642-02258-6 ), Chapter 16