Carathéodory's theorem

In Euclidean space of dimension d, every point in the convex hull of a set is a convex combination of at most d+1 points from that set.

Affine dependence: A set of points is affinely dependent when some nonzero coefficients sum to zero and their weighted point sum is zero. An affine dependence lets one adjust coefficients until a point drops out without changing the combination.
Helly's theorem: A convexity theorem stating that, under dimension-dependent conditions, a family of convex sets intersects if every small subfamily intersects. Its standard proof applies Carathéodory's theorem to separating convex combinations.
Convex set: A set containing the line segment between every pair of its points. Convex hulls and convex combinations are defined through this closure property.
Constantin Carathéodory: A Greek-German mathematician whose work helped establish modern convex geometry. He proved the theorem that bears his name.
Convex hull: The smallest convex set containing a given set, equivalently all its finite convex combinations. The theorem characterizes every point in this set using a bounded number of generators.
Tverberg's theorem: A theorem guaranteeing a partition of sufficiently many points into groups whose convex hulls share a point. Its proof repeatedly uses the bounded-support representations that Carathéodory provides.
Euclidean space: A real vector space equipped with the standard inner product and its associated distance. The theorem's dimension d is the dimension of the Euclidean space under consideration.
Convex geometry: The study of convex sets and their geometric and combinatorial properties. Carathéodory's theorem is a foundational result in this field.
Affine span: The smallest affine subspace containing a set of points. The sharper bound uses the dimension of the affine span rather than the ambient space.
Linear programming: Optimization of a linear objective over variables satisfying linear constraints. Extreme-point solutions and basic feasible solutions reflect dimension-bounded convex representations.

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