The Borsuk problem in geometry, for historical reasons [note 1] incorrectly called Borsuk's conjecture , is a question in discrete geometry. It is named after Karol Borsuk.
The Borsuk problem in geometry, for historical reasons [note 1] incorrectly called Borsuk's conjecture , is a question in discrete geometry. It is named after Karol Borsuk.
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. At the same time he proved that n subsets are not enough in general. The proof is based on the Borsuk–Ulam theorem. 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 be partitioned into (n + 1) sets, each of which has a smaller diameter than E?
— Drei Sätze über die n-dimensionale euklidische Sphäre
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 many pieces. 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. [16] However, it is conjectured that there is a constant c > 1 such that α(n) > cn for all n ≥ 1.
Oded Schramm also worked in a related question, a body of constant width is said to have effective radius if , where is the unit ball in , he proved the lower bound , where is the smallest effective radius of a body of constant width 2 in and asked if there exists such that for all , [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 . [19] [20] [21]
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