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 ge ...

, 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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

can be partitioned into

subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset o ...

s with intersecting convex hulls. Specifically, for any set of :

(d + 1)(r - 1) + 1\

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 as Radon's theorem. 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 In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of '' septua-'', a Latin-derived numerical prefix, rather than '' hepta-'', a Greek-derived n ...

. 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.

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

Examples

See also

References