Borsuk's conjecture

Borsuk's conjecture is a mathematical conjecture in the field of geometry. It is about the question must be broken down into as many parts to a given amount of limited diameter, so that each part has a really smaller diameter. The 1933 by Karol Borsuk asked and later designated as a conjecture whether one always gets along with parts in dimensions, was answered 60 years later negative.

The assumption

In the n- dimensional space can be defined as (maximum distance between two points of the set ) by means of the Euclidean norm, the diameter of a set.

One can try to disassemble the set into subsets so now that everyone has a real part as smaller diameter. This raises the question of how many subsets to be required.

As the regular n-dimensional simplex is at least amounts are generally necessary, because the corners are all of the same pitch which is equal to the diameter. A subset of real smaller diameter may therefore contain at most one corner, that is, one needs at least as many subsets as there are corners, and it has one. How to make adjacent drawing for the dimensions 1,2 and 3 clearly one actually comes with the simplex subsets from. Karol Borsuk completed his work "Three theorems on n- dimensional sphere ", in which he dealt with the separation of spheres, as follows:

The assumption that this question should be answered in the affirmative, was known as Borsuk 's conjecture and remained open for 60 years.

Refutation

In the space of intuition, the presumption was confirmed in 1955. It may be surprising, therefore, that the Borsuk conjecture proves false in higher dimensions. 1993 Jeff Kahn and G. Kalai have shown that at least part quantities required for sufficiently large dimensions, so that the Borsuk 's conjecture was refuted, because growing faster than. A concrete counterexample was found by A. Nilli in the 964 -dimensional space, and later another A. Hinrichs and C. Richter in the 298- dimensional space. Today it is known that the Borsuk 's conjecture is false for dimensions from 64. The question of the smallest dimension, from which the Borsuk conjecture is no longer the case is open.