Borsuk's conjecture

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

The Borsuk conjecture is a claim in discrete geometry, now disproved. It states that:

Every convex body in $\Bbb R^d$ can be cut into $(d + 1)$ pieces of smaller diameter.

The conjecture was introduced in 1932 by Karol Borsuk, who showed that d pieces are not enough for a ball. The proof is based on the Borsuk–Ulam theorem.

The conjecture was established in the following cases:

d=2 – the original result by Borsuk (1932).
d=3 – the result of H. G. Eggleston (1955). A simple proof was found later by Branko Grünbaum and Aladár Heppes.
For all d for the smooth convex bodies – the result of Hugo Hadwiger (1946).
For all d for centrally-symmetric bodies.
For all d for bodies of revolution – the result of Dexter (1995).

The conjecture was disproved in 1993 by Jeff Kahn and Gil Kalai. The current best bound, due to Aicke Hinrichs and Christian Richter, shows that the conjecture is false for all d ≥ 298. The proof by Kahn and Kalai implies that for large enough d, one needs $\alpha(d) > c^\sqrt{d}$ number of pieces. It is conjectured that (see e.g. Alon's article) that $α(d) > c d$ for some c >1.