Tverberg's theorem

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. Specifically, for any positive integers

d

,

r

{\displaystyle d,r}

and any set of

(

d

+

1

)

(

r

−

1

)

+

1

{\displaystyle (d+1)(r-1)+1\ }

points in

d

{\displaystyle d}

-dimensional Euclidean space there exists a partition of the given points into

r

{\displaystyle r}

subsets whose convex hulls all have a common point; in other words, there exists a point

x

{\displaystyle x}

(not necessarily one of the given points) such that

x

{\displaystyle x}

belongs to the convex hull of all of the subsets.

The partition resulting from this theorem is known as a Tverberg partition.

The special case

r

=

2

{\displaystyle r=2}

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

Examples

The case

d

=

1

{\displaystyle d=1}

states that any

2

r

−

1

{\displaystyle 2r-1}

points on the real line can be partitioned into

r

{\displaystyle r}

subsets with intersecting convex hulls. Indeed, if the points are

x

1

<

x

2

<

.

.

.

<

x

2

r

−

1

{\displaystyle x_{1}<x_{2}<...<x_{2r-1}}

, then the partition into

A

i

=

{

x

i

,

x

2

r

−

i

}

{\displaystyle A_{i}=\{x_{i},x_{2r-i}\}}

for

i

=

1

,

.

.