In discrete geometry, Tverberg's theorem, first stated by Helge Tverberg in 1966, is the result that sufficiently many points in d-dimensional Euclidean space can be partitioned into subsets with intersecting convex hulls. Specifically, for any positive integers d, r and any set of
(d+1)(r-1)+1
The special case r = 2 was proved earlier by Radon, and it is known as Radon's theorem.
The case d = 1 states that any 2r−1 points on the real line can be partitioned into r subsets with intersecting convex hulls. Indeed, if the points are x1 < x2 < ... < x2r < x2r-1, then the partition into Ai = for i in 1,...,r satisfies this condition (and it is unique).
For r = 2, Tverberg's theorem states that any d + 2 points 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.
The case r = 3 and d = 2 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.
An equivalent formulation of Tverberg's theorem is:
Let d, r be positive integers, and let N := (d+1)(r-1). If ƒ is any affine function from an N-dimensional simplex ΔN to Rd, then there are r pairwise-disjoint faces of ΔN whose images under ƒ intersect. That is: there exist faces F1,...,Fr of ΔN such thatThey are equivalent because any affine function on a simplex is uniquely determined by the images of its vertices. Formally, let ƒ be an affine function from ΔN to Rd. Letand\foralli,j\in[r]:Fi\capFj=\emptyset
.f(F1)\cap … \capf(Fr) ≠ \emptyset
v1,v2,...,vN+1
x1,x2,...,xN+1
x1,x2,...,xN+1
The topological Tverberg theorem generalizes this formluation. It allows f to be any continuous function - not necessarily affine. But, currently it is proved only for the case where r is a prime power:
Let d be a positive integer, and let r be a power of a prime number. Let N := (d+1)(r-1). If ƒ is any continuous function from an N-dimensional simplex ΔN to Rd, then there are r pairwise-disjoint faces of ΔN whose images under ƒ intersect. That is: there exist faces F1,...,Fr of ΔN such thatand\foralli,j\in[r]:Fi\capFj=\emptyset
.f(F1)\cap … \capf(Fr) ≠ \emptyset
The topological Tverberg theorem was proved for prime
r
The theorem was proved for
r