Tverberg's theorem

For integers r ≥ 2 and d ≥ 1, every set of (r−1)(d+1)+1 points in ℝᵈ can be partitioned into r groups whose convex hulls share a point.

Convex hull: The smallest convex set containing a given set of points. Each group in the theorem contributes one convex hull to the common intersection.
Tverberg partition: A partition of a point set into groups whose convex hulls have a common point. It is the concrete structure guaranteed by the theorem.
Helge Tverberg: Norwegian mathematician known for the convexity theorem that bears his name. He proved the theorem in 1966.
Centerpoint theorem: A theorem guaranteeing a point that lies in a substantial fraction of every halfspace containing it. Tverberg partitions yield centerpoint bounds by taking a common point of group hulls.
Topological Tverberg conjecture: The proposed extension of the topological Tverberg theorem to all relevant dimensions and multiplicities. Unlike the affine theorem, this extension fails in some dimensions and remains delicate in others.
Convex set: A set containing the entire line segment between any two of its points. The theorem concerns intersections of convex sets formed from point groups.
Barycentric coordinates: Coordinates expressing a point as an affine combination of vertices in a simplex. They describe the convex combinations that place one point in every group’s hull.
Convex geometry: The study of convex sets, their geometric structure, and their combinatorial properties. Tverberg’s theorem is a central result in this field.
Weak ε-net: A finite set intersecting every convex set that contains a specified positive fraction of a point set. Tverberg partitions help construct small weak nets for convex ranges.
Van Kampen–Flores theorem: A nonembedding theorem for skeleta of simplices in Euclidean spaces. It is a related topological obstruction, not a convex-hull partition guarantee.

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