Borsuk's conjecture

File:Borsuk Hexagon.svg

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 Template:Mvar of the space ${\mathbb{ℝ}}^n$ can be partitioned into (n + 1) sets, each of which has a smaller diameter than Template:Mvar?

More unsolved problems in mathematics

The Borsuk problem in geometry, for historical reasonsTemplate:Refn incorrectly called Borsuk's conjecture, is a question in discrete geometry. It is named after Karol Borsuk.

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 Template:Mvar-dimensional ball can be covered with n + 1 compact sets of diameters smaller than the ball. At the same time he proved that Template:Mvar subsets are not enough in general. The proof is based on the Borsuk–Ulam theorem. That led Borsuk to a general question:

The following question remains open: Can every bounded subset Template:Mvar of the space ${\mathbb{ℝ}}^n$ be partitioned into (n + 1) sets, each of which has a smaller diameter than Template:Mvar?

— Template:Comma separated entries

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), and independently, 8 years later, by H. G. Eggleston (1955). A simple proof was found later by Branko Grünbaum and Aladár Heppes.
• For all Template:Mvar for smooth convex fields — shown by Hugo Hadwiger (1946).
• For all Template:Mvar for centrally-symmetric fields — shown by A.S. Riesling (1971).
• For all Template:Mvar for fields of revolution — shown by Boris Dekster (1995).

The problem was finally solved in 1993 by Jeff Kahn and Gil Kalai, who showed that the general answer to Borsuk's question is Template:Em. 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, 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).

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.

In 2013, Andriy V. Bondarenko had shown that Borsuk's conjecture is false for all n ≥ 65. Shortly after, Thomas Jenrich derived a 64-dimensional counterexample from Bondarenko's construction, giving the best bound up to now.

Apart from finding the minimum number Template:Mvar 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 Template:Mvar sufficiently large), one needs $\alpha (n)\ge (1.2{)}^{\sqrt{n}}$ many pieces. They also quote the upper bound by Oded Schramm, who showed that for every Template:Mvar, if Template:Mvar is sufficiently large, $\alpha (n)\le {(\sqrt{3/2}+\epsilon )}^n$. The correct order of magnitude of α(n) is still unknown. 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 $K$ of constant width is said to have effective radius $r$ if $\text{Vol}(K)=r^n\text{Vol}({\mathbb{𝔹}}^n)$, where ${\mathbb{𝔹}}^n$ is the unit ball in ${\mathbb{ℝ}}^n$, he proved the lower bound $\sqrt{3+2/(n+1)}-1\le r_n$, where $r_n$ is the smallest effective radius of a body of constant width 2 in ${\mathbb{ℝ}}^n$ and asked if there exists $ϵ>0$ such that $r_n\le 1-ϵ$ for all $n\ge 2$, 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 $\text{Vol}(K)\le (0.9{)}^n\text{Vol}({\mathbb{𝔹}}^n)$.

See also

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

Further reading

• Oleg Pikhurko, Algebraic Methods in Combinatorics, course notes.
• Andrei M. Raigorodskii, The Borsuk partition problem: the seventieth anniversary, Mathematical Intelligencer 26 (2004), no. 3, 4–12.