Borsuk's conjecture

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.

An example of a hexagon cut into three pieces of smaller diameter.

Unsolved problem in mathematics

What is the lowest n such that not every bounded subset E of the space ${\displaystyle \mathbb {R} ^{n}}$ can be partitioned into (n + 1) sets, each of which has a smaller diameter than E?

More unsolved problems in mathematics

Problem

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 ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle \mathbb {R} ^{n}}$ 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:

• n = 2 — which is the original result by Karol Borsuk (1932).
• n = 3 — shown by Julian Perkal (1947),^[3] and independently, 8 years later, by H. G. Eggleston (1955).^[4] A simple proof was found later by Branko Grünbaum and Aladár Heppes.
• For all n for smooth convex fields — shown by Hugo Hadwiger (1946).^[5][6]
• For all n for centrally-symmetric fields — shown by A.S. Riesling (1971).^[7]
• For all n for fields of revolution — shown by Boris Dekster (1995).^[8]

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 ${\textstyle \alpha (n)\geq (1.2)^{\sqrt {n}}}$ many pieces. They also quote the upper bound by Oded Schramm, who showed that for every ε, if n is sufficiently large, ${\textstyle \alpha (n)\leq \left({\sqrt {3/2}}+\varepsilon \right)^{n}}$.^[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 ${\displaystyle K}$ of constant width is said to have effective radius ${\displaystyle r}$ if ${\displaystyle {\text{Vol}}(K)=r^{n}{\text{Vol}}(\mathbb {B} ^{n})}$, where ${\displaystyle \mathbb {B} ^{n}}$ is the unit ball in ${\displaystyle \mathbb {R} ^{n}}$, he proved the lower bound ${\displaystyle {\sqrt {3+2/(n+1)}}-1\leq r_{n}}$, where ${\displaystyle r_{n}}$ is the smallest effective radius of a body of constant width 2 in ${\displaystyle \mathbb {R} ^{n}}$ and asked if there exists ${\displaystyle \epsilon >0}$ such that ${\displaystyle r_{n}\leq 1-\epsilon }$ for all ${\displaystyle n\geq 2}$,^[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 ${\displaystyle {\text{Vol}}(K)\leq (0.9)^{n}{\text{Vol}}(\mathbb {B} ^{n})}$.^[19][20][21]

See also

• Hadwiger's conjecture on covering convex fields with smaller copies of themselves
• Kahn–Kalai conjecture

Note

1. As Hinrichs and Richter say in the introduction to their work,^[1] the "Borsuk's conjecture [was] believed by many to be true for some decades" (hence commonly called a conjecture) so "it came as a surprise when Kahn and Kalai constructed finite sets showing the contrary". However, Karol Borsuk has formulated the problem just as a question, not suggesting the expected answer would be positive.