Borsuk's conjecture explained

The Borsuk problem in geometry, for historical reasons 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 -dimensional ball can be covered with compact sets of diameters smaller than the ball. At the same time he proved that subsets are not enough in general. The proof is based on the Borsuk–Ulam theorem. That led Borsuk to a general question:

The question was answered in the positive in the following cases:

• — which is the original result by Karol Borsuk (1932).
• — 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 for smooth convex fields — shown by Hugo Hadwiger (1946).
• For all for centrally-symmetric fields — shown by A.S. Riesling (1971).
• For all 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 . They claim that their construction shows that pieces do not suffice for and for each . 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 (as well as all higher dimensions up to 1560).

Their result was improved in 2003 by Hinrichs and Richter, who constructed finite sets for, which cannot be partitioned into parts of smaller diameter.

In 2013, Andriy V. Bondarenko had shown that Borsuk's conjecture is false for all . 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 of dimensions such that the number of pieces, mathematicians are interested in finding the general behavior of the function . Kahn and Kalai show that in general (that is, for sufficiently large), one needs $\backslash alpha(n)\; \backslash ge\; (1.2)^\backslash sqrt$ many pieces. They also quote the upper bound by Oded Schramm, who showed that for every, if is sufficiently large, $\backslash alpha(n)\; \backslash le\; \backslash left(\backslash sqrt\; +\; \backslash varepsilon\backslash right)^n$. The correct order of magnitude of is still unknown. However, it is conjectured that there is a constant such that for all .

Oded Schramm also worked in a related question, a body

K of constant width is said to have effective radius r if Vol (K) = r ^{nVol(B} ^{n} ) , where B ^{n} is the unit ball in 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 R ^{n} and asked if there exists \epsilon> 0 such that r _{n\le} 1- \epsilon for all n\ge 2 ,^{[1] [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 Vol (K)\leq ( 0 . 9 ) ^{nVol(B} ^{n} ) .^{[3] [4] [5]}

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.
• Book: Raigorodskii, Andreii M. . Three lectures on the Borsuk partition problem . 1144.52005 . Young . Nicholas . Choi . Yemon . Surveys in contemporary mathematics . . 978-0-521-70564-6 . London Mathematical Society Lecture Note Series . 347 . 202–247 . 2008 .

Notes and References

1. Schramm . Oded . June 1988 . On the volume of sets having constant width . Israel Journal of Mathematics . en . 63 . 2 . 178–182 . 10.1007/BF02765037 . 0021-2172.
2. Kalai . Gil . Some old and new problems in combinatorial geometry I: Around Borsuk's problem . 2015-05-19 . math.CO . 1505.04952.
3. Arman . Andrii . Small volume bodies of constant width . 2024-05-28 . 2405.18501 . Bondarenko . Andriy . Nazarov . Fedor . Prymak . Andriy . Radchenko . Danylo. math.MG .
4. Web site: Kalai . Gil . 2024-05-31 . Andrii Arman, Andriy Bondarenko, Fedor Nazarov, Andriy Prymak, and Danylo Radchenko Constructed Small Volume Bodies of Constant Width . 2024-09-28 . Combinatorics and more . en.
5. Web site: Barber . Gregory . 2024-09-20 . Mathematicians Discover New Shapes to Solve Decades-Old Geometry Problem . 2024-09-28 . Quanta Magazine . en.