Borsuk's conjecture

File:Borsuk Hexagon.svg

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

The Borsuk problem in geometry, for historical reasons incorrectly called Borsuk's conjecture, is a question in discrete geometry.

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

Die folgende Frage bleibt offen: Lässt sich jede beschränkte Teilmenge E des Raumes $\Bbb R^n$ in (n + 1) Mengen zerlegen, von denen jede einen kleineren Durchmesser als E hat?

This can be translated as:

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

The question got a positive answer in the following cases:

• d = 2 — which is the original result by Karol Borsuk (1932).
• d = 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 d for the smooth convex bodies — shown by Hugo Hadwiger (1946).
• For all d for centrally-symmetric bodies — shown by A.S. Riesling (1971).
• For all d for bodies 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 no. Their construction shows that d + 1 pieces do not suffice for d = 1,325 and for each d > 2,014.

After Andriy V. Bondarenko had shown that Borsuk’s conjecture is false for all d ≥ 65, the current best bound, due to Thomas Jenrich, is 64.

Apart from finding the minimum number d of dimensions such that the number of pieces $\alpha(d) > d+1$ mathematicians are interested in finding the general behavior of the function $\alpha(d)$. Kahn and Kalai show that in general (that is for d big enough), one needs $\alpha(d) \ge (1.2)^\sqrt{d}$ number of pieces. They also quote the upper bound by Oded Schramm, who showed that for every ε, if d is sufficiently large, $\alpha(d) \le \left(\sqrt{3/2} + \varepsilon\right)^d$. The correct order of magnitude of α(d) is still unknown. However, it is conjectured that there is a constant c > 1 such that $\alpha(d) > c^d$ for all d ≥ 1.

See also

• Hadwiger's conjecture on covering convex bodies with smaller copies of themselves

Further reading

• Oleg Pikhurko, Algebraic Methods in Combinatorics, course notes.
• Aicke Hinrichs and Christian Richter, New sets with large Borsuk numbers, Discrete Mathematics 270 (2003), 137–147
• Andrei M. Raigorodskii, The Borsuk partition problem: the seventieth anniversary, Mathematical Intelligencer 26 (2004), no. 3, 4–12.

External links

• Weisstein, Eric W., "Borsuk's Conjecture", MathWorld.