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In discrete geometry, Tverberg's theorem, first stated by , is the result that sufficiently many points in ''d''-dimensional
In discrete geometry, Tverberg's theorem, first stated by , is the result that sufficiently many points in ''d''-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
can be partitioned into subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s with intersecting convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
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 all 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 asRadon's theorem
In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that any set of ''d'' + 2 points in R''d'' can be partitioned into two sets whose convex hulls intersect. A point in the intersection of these conve ...
. 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.
See also
*Rota's basis conjecture
In linear algebra and matroid theory, Rota's basis conjecture is an unproven conjecture concerning rearrangements of bases, named after Gian-Carlo Rota. It states that, if ''X'' is either a vector space of dimension ''n'' or more generally a matr ...
References
*. *{{citation , last = Hell , first = S. , publisher = Dissertation, TU Berlin , title = Tverberg-type theorems and the Fractional Helly property , year = 2006, doi = 10.14279/depositonce-1464 . Theorems in convex geometry Theorems in discrete geometry Geometric transversal theory Convex hulls