Helly's theorem

Helly's theorem states that a finite family of convex sets in Euclidean space has nonempty intersection if every subfamily of at most one more than the dimension intersects.

Convex set: A set containing the line segment between every pair of its points. Convexity is the geometric condition that makes the intersection criterion work.
Linear programming: Optimization of a linear objective over a region defined by linear constraints. Helly-type arguments yield bounds on infeasible constraint subsystems.
Eduard Helly: An Austrian mathematician whose work established the finite-dimensional intersection theorem bearing his name. His 1913 work introduced the result now called Helly's theorem.
Infinite-dimensional Helly theorem: A family of convex sets in an infinite-dimensional space generally needs additional compactness conditions for a Helly-type conclusion. The finite-dimensional theorem does not transfer unchanged to infinite-dimensional spaces.
Radon's theorem: Any set of d + 2 points in d-dimensional Euclidean space can be partitioned into two parts whose convex hulls intersect. Radon's partition lemma provides a standard route to proving Helly's theorem.
Helly property: A property in which a family has a common intersection whenever all sufficiently small subfamilies intersect. This abstracts the finite intersection principle beyond Euclidean convex sets.
Hans Rademacher: A German-American mathematician known for work in analysis, number theory, and geometry. He and R. E. Jamison helped popularize the theorem through a later proof.
Topological Helly theorem: A family-intersection theorem extending Helly's principle to suitable topological spaces and acyclic sets. It replaces Euclidean convexity with topological conditions on intersections.
Carathéodory's theorem: A point in the convex hull of a set in d-dimensional space lies in the convex hull of at most d + 1 of its points. It shares the dimension threshold and underlies related convexity arguments.
Linear inequalities: Constraints that compare linear expressions using relations such as ≤ or ≥. Half-spaces defined by inequalities let Helly's theorem test feasibility.

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