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The Borsuk problem in geometry, for historical reasons incorrectly called Borsuk's conjecture, is a question in discrete geometry.
Problem
In 1932 Karol Borsuk showed that an ordinary 3-dimensional ball in Euclidean space can be easily dissected into 4 solids, each of which has a smaller diameter than the ball, and generally d-dimensional ball can be covered with d + 1 compact sets of diameters smaller than the ball. At the same time he proved that d subsets are not enough in general. The proof is based on the Borsuk–Ulam theorem. That led Borsuk to a general question:
Die folgende Frage bleibt offen: Lässt sich jede beschränkte Teilmenge E des RaumesThis can be translated as:
The following question remains open: Can every bounded subset E of the spaceThe question got a positive answer in the following cases:
The problem was finally solved in 1993 by Jeff Kahn and Gil Kalai, who showed that the general answer to Borsuk's question is no. Their construction shows that d + 1 pieces do not suffice for d = 1,325 and for each d > 2,014.
After Andriy V. Bondarenko had shown that Borsuk’s conjecture is false for all d ≥ 65, the current best bound, due to Thomas Jenrich, is 64.
Apart from finding the minimum number d of dimensions such that the number of pieces