Helly's theorem
/ HEL-ee /
You have a big collection of convex shapes, and you want to know whether they all share a common point. Checking the whole pile at once is hard. Helly's theorem gives a stunning shortcut in n dimensions: you only have to check that every n + 1 of them have a common point. If every little group of n + 1 shapes overlaps, then magically all of them overlap simultaneously. Local agreement among small subfamilies forces global agreement.
Precisely, let C_1, ..., C_m be convex sets in R^n with m >= n + 1. Helly's theorem states: if every n + 1 of these sets have a nonempty common intersection, then all m of them have a nonempty common intersection. (For infinitely many sets the same holds provided the sets are compact, to supply a finite-intersection-property compactness argument.) In the plane (n = 2) the magic number is 3: if every three of your convex sets meet, they all meet. The proof rests on Radon's theorem — partition n + 2 points into two groups whose convex hulls intersect — applied to cleverly chosen witness points, an elegant induction.
Helly's theorem is the cornerstone of combinatorial convexity and seeds a whole industry of 'Helly-type' results in geometry, optimization, and even data science. It explains why many feasibility problems reduce to checking small subsystems, underlies centerpoint and ham-sandwich type theorems, and gives clean proofs that, e.g., a family of intervals on a line pairwise overlapping must share a common point (the n = 1 case). Two honest caveats. First, convexity is indispensable: for non-convex sets the theorem fails outright — three crescents can pairwise overlap with no common point. Second, the number n + 1 is exactly right and cannot be lowered; in the plane 'every two meet' does not force a common point (three long thin triangles around a small hole pairwise cross but share nothing).
On a line (n = 1) Helly's number is 2. Take any family of intervals such that every two of them overlap; Helly says they all share a common point. Concretely, intervals [0,3], [1,4], [2,5] overlap pairwise, and indeed [2,3] is common to all three. The theorem promises this is no coincidence: pairwise overlap of intervals always forces a global common point — which fails the moment you allow non-convex sets like pairs of disjoint segments.
On a line, pairwise-overlapping intervals always share a common point (Helly, n=1).
The threshold n + 1 cannot be reduced and convexity cannot be dropped. Both are tight: in the plane three thin triangles can meet pairwise around a hole with empty triple intersection, and any non-convex example breaks the conclusion even with full pairwise overlap.