The Borsuk problem in geometry, for historical reasons incorrectly called Borsuk's

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

, is a question in

discrete geometry Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geom ...

. It is named after Karol Borsuk.

Problem

In 1932, Karol Borsuk showed that an ordinary 3-dimensional

ball A ball is a round object (usually spherical, but sometimes ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for s ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

can be easily dissected into 4 solids, each of which has a smaller

diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

than the ball, and generally -dimensional ball can be covered with

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

sets of diameters smaller than the ball. At the same time he proved that

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

s 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 Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentsmooth convex fields — shown by

Hugo Hadwiger Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss people, Swiss mathematician, known for his work in geometry, combinatorics, and cryptography. Biography Although born in Karlsruhe, Ge ...

(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

\alpha(n) \ge (1.2)^\sqrt

many pieces. They also quote the upper bound by Oded Schramm, who showed that for every , if is sufficiently large,

\alpha(n) \le \left(\sqrt + \varepsilon\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

\text(K)=r^n\text(\mathbb^ )

, where

\mathbb^

is the unit ball in

\mathbb^

, he proved the lower bound

\sqrt-1\le r_n

, where

r_n

is the smallest effective radius of a body of constant width 2 in

\mathbb^

and asked if there exists

\epsilon>0

such that

r_n\le 1-\epsilon

for all

n\ge2

, 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 Nazarov (), or Nazarova (feminine; Назарова) is a Russian family name. The surname derives from the given name Nazar (given name), Nazar. The surname may refer to: *Alexander Nazarov (1925–1945), Soviet army officer and Hero of the Sovie ...

, Prymak, Radchenko reported to have answered this question in the affirmative giving a construction that satisfies

\text(K)\leq (0.9)^n\text(\mathbb^ )

Note

References

External links

* {{MathWorld, urlname=BorsuksConjecture, title=Borsuk's Conjecture Disproved conjectures Discrete geometry

Problem

See also

Note

References

Further reading

External links