Tverberg's theorem
Encyclopedia
In discrete geometry
, Tverberg's theorem, first stated by , is the result that sufficiently many points in d-dimensional Euclidean space
can be partitioned
into subsets with intersecting convex hull
s. Specifically, for any set of
points there exists a point x (not necessarily one of the given points) and a partition of the given points into r subsets, such that x belongs to the convex hull of each of the subsets. The partition resulting from this theorem is known as a Tverberg partition.
. In this case, for points in general position, there is a unique partition.
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.
Discrete geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles,...
, Tverberg's theorem, first stated by , is the result that sufficiently many points in d-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
can be partitioned
Partition of a set
In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...
into subsets with intersecting convex hull
Convex hull
In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
s. Specifically, for any set of
points there exists a point x (not necessarily one of the given points) and a partition of the given points into r subsets, such that x belongs to the convex hull of each of the subsets. The partition resulting from this theorem is known as a Tverberg partition.
Examples
For r = 2, Tverberg's theorem states that any d + 2 points may be partitioned into two subsets with intersecting convex hulls; this special case is known as Radon's theoremRadon's theorem
In geometry, Radon's theorem on convex sets, named after Johann Radon, states that any set of d + 2 points in Rd can be partitioned into two sets whose convex hulls intersect...
. In this case, for points in general position, there is a unique partition.
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.