In discrete geometry, Tverberg's theorem, first stated by Helge Tverberg in 1966, is the result that sufficiently many points in Euclidean space can be partitioned into subsets with intersecting convex hulls and any set of

-dimensional Euclidean space belongs to the convex hull of all of the subsets. The partition resulting from this theorem is known as a Tverberg partition.

was proved earlier by Radon, and it is known as Radon's theorem.

Examples

satisfies this condition (and it is unique).

-dimensional Euclidean space may be partitioned into two subsets with intersecting convex hulls. This is known as Radon's theorem. In this case, for points in general position, the partition is unique.

states that any seven points in the plane may be partitioned into three subsets with intersecting convex hulls. The illustration shows an example in which the seven points are the vertices of a regular heptagon. As the example shows, there may be many different Tverberg partitions of the same set of points; these seven points may be partitioned in seven different ways that differ by rotations of each other.

Topological Tverberg Theorem

An equivalent formulation of Tverberg's theorem is the following: is any affine function-dimensional simplex. be an affine function intersect, as claimed by the reformulation.

The topological Tverberg theorem (first hypothesized by Bárány in a 1976 letter to Tverberg is a prime power: is any continuous function-dimensional simplex.

Proofs and Refutations

by Bárány, Shlosman and Szűcs. Matoušek presents a proof using deleted joins.

a prime-power by Özaydin, and later by Volovikov and Sarkaria.

It was a long-standing open problem. However, in 2015 Frick-fold Whitney trick" by Mabillard and Wagner, and the "constraint method" by Blagojević, Ziegler and Frick leads to counterexamples.

See also

• Rota's basis conjecture

References

Further reading