Borsuk's conjecture

The Borsuk problem in geometry, for historical reasons incorrectly called Borsuk's conjecture, is a question in discrete geometry.

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:

This can be translated as:

The 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 ${\displaystyle \alpha (d)>d+1}$ mathematicians are interested in finding the general behavior of the function ${\displaystyle \alpha (d)}$. Kahn and Kalai show that in general (that is for d big enough), one needs ${\displaystyle \alpha (d)\geq (1.2)^{\sqrt {d}}}$ number of pieces. They also quote the upper bound by Oded Schramm, who showed that for every ε, if d is sufficiently large, ${\displaystyle \alpha (d)\leq \left({\sqrt {3/2}}+\varepsilon \right)^{d}}$. The correct order of magnitude of α(d) is still unknown. However, it is conjectured that there is a constant c > 1 such that ${\displaystyle \alpha (d)>c^{d}}$ for all d ≥ 1.

...
Wikipedia