Radon's theorem

Radon's theorem states that any set of at least d + 2 points in d-dimensional Euclidean space can be partitioned into two subsets whose convex hulls intersect.

Affine dependence: A linear dependence among points with coefficients summing to zero. Its positive and negative coefficients divide the points into groups with equal weighted averages.
Tverberg's theorem: A theorem guaranteeing a partition of points into several groups whose convex hulls share a point. For two groups, its dimension-dependent point bound extends Radon's guarantee.
Johann Radon: An Austrian mathematician whose work included measure theory, integral geometry, and convexity. The theorem bears his name and appeared in his 1921 work on convex sets.
Radon point: A point lying in the convex hulls of both parts of a Radon partition. It is the intersection point guaranteed by the theorem.
Convex hull: The smallest convex set containing a given set of points. The theorem asserts that the two groups' convex hulls share a point.
Helly's theorem: A convexity theorem stating that finite families of convex sets intersect if every small subfamily intersects. Radon's partition argument is a standard route to Helly's theorem.
Convex geometry: The study of convex sets and their geometric and combinatorial properties. Radon's theorem became a foundational result in this field.
Radon partition: A partition of a finite point set into two parts whose convex hulls intersect. This is the specific partition whose existence Radon's theorem guarantees.
Barycentric coordinates: Coordinates that express a point as an affine combination of the vertices of a simplex. The shared point is expressed as a convex combination from each side of the partition.
Kirchberger's theorem: A theorem characterizing whether two finite point sets can be strictly separated by a hyperplane. Radon's theorem supplies the obstruction used in its separation criterion.

Linked from 4 pages

Shapley–Folkman lemmaCompared with: It yields convexity structure through point partitions, not approximation of sums.