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.