Borsuk's conjecture
The Borsuk problem in geometry, for historical reasons[note 1] incorrectly called Borsuk's conjecture, is a question in discrete geometry.
In 1932, Karol Borsuk showed[2] 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 n-dimensional ball can be covered with n + 1 compact sets of diameters smaller than the ball.
That led Borsuk to a general question:[2] Die folgende Frage bleibt offen: Lässt sich jede beschränkte Teilmenge E des Raumes
in (n + 1) Mengen zerlegen, von denen jede einen kleineren Durchmesser als E hat?
The following question remains open: Can every bounded subset E of the space
The question was answered in the positive 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.
[9] They claim that their construction shows that n + 1 pieces do not suffice for n = 1325 and for each n > 2014.
However, as pointed out by Bernulf Weißbach,[10] the first part of this claim is in fact false.
But after improving a suboptimal conclusion within the corresponding derivation, one can indeed verify one of the constructed point sets as a counterexample for n = 1325 (as well as all higher dimensions up to 1560).
[11] Their result was improved in 2003 by Hinrichs and Richter, who constructed finite sets for n ≥ 298, which cannot be partitioned into n + 11 parts of smaller diameter.
[1] In 2013, Andriy V. Bondarenko had shown that Borsuk's conjecture is false for all n ≥ 65.
[12] Shortly after, Thomas Jenrich derived a 64-dimensional counterexample from Bondarenko's construction, giving the best bound up to now.
[13][14] Apart from finding the minimum number n of dimensions such that the number of pieces α(n) > n + 1, mathematicians are interested in finding the general behavior of the function α(n).
Kahn and Kalai show that in general (that is, for n sufficiently large), one needs
They also quote the upper bound by Oded Schramm, who showed that for every ε, if n is sufficiently large,
[15] The correct order of magnitude of α(n) is still unknown.
Oded Schramm also worked in a related question, a body
is the smallest effective radius of a body of constant width 2 in
,[17][18] that is if the gap between the volumes of the smallest and largest constant-width bodies grows exponentially.
In 2024 a preprint by Arman, Bondarenko, Nazarov, Prymak, Radchenko reported to have answered this question in the affirmative giving a construction that satisfies
