Helly's theorem is a basic result in

discrete geometry Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic ge ...

on the intersection of

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...

s. It was discovered by

Eduard Helly Eduard Helly (June 1, 1884 in Vienna – 28 November 1943 in Chicago) was a mathematician after whom Helly's theorem, Helly families, Helly's selection theorem, Helly metric, and the Helly–Bray theorem were named. Life Helly earned his doct ...

in 1913,. but not published by him until 1923, by which time alternative proofs by and had already appeared. Helly's theorem gave rise to the notion of a

Helly family In combinatorics, a Helly family of order is a family of sets in which every minimal ''subfamily with an empty intersection'' has or fewer sets in it. Equivalently, every finite subfamily such that every -fold intersection is non-empty has non ...

Statement

Let be a finite collection of convex subsets of , with . If the intersection of every of these sets is nonempty, then the whole collection has a nonempty intersection; that is, :

\bigcap_^n X_j\ne\varnothing.

For infinite collections one has to assume compactness: Let be a collection of

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

convex subsets of , such that every subcollection of cardinality at most has nonempty intersection. Then the whole collection has nonempty intersection.

Proof

We prove the finite version, using Radon's theorem as in the proof by . The infinite version then follows by the

finite intersection property In general topology, a branch of mathematics, a non-empty family ''A'' of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is non-empty. It has the strong finite inters ...

characterization of compactness: a collection of closed subsets of a compact space has a non-empty intersection if and only if every finite subcollection has a non-empty intersection (once you fix a single set, the intersection of all others with it are closed subsets of a fixed compact space). The proof is by induction: Base case: Let . By our assumptions, for every there is a point that is in the common intersection of all with the possible exception of . Now we apply Radon's theorem to the set which furnishes us with disjoint subsets of such that the convex hull of intersects the convex hull of . Suppose that is a point in the intersection of these two convex hulls. We claim that :

p\in\bigcap_^n X_j.

Indeed, consider any We shall prove that Note that the only element of that may not be in is . If , then , and therefore . Since is convex, it then also contains the convex hull of and therefore also . Likewise, if , then , and by the same reasoning . Since is in every , it must also be in the intersection. Above, we have assumed that the points are all distinct. If this is not the case, say for some , then is in every one of the sets , and again we conclude that the intersection is nonempty. This completes the proof in the case . Inductive Step: Suppose and that the statement is true for . The argument above shows that any subcollection of sets will have nonempty intersection. We may then consider the collection where we replace the two sets and with the single set . In this new collection, every subcollection of sets will have nonempty intersection. The inductive hypothesis therefore applies, and shows that this new collection has nonempty intersection. This implies the same for the original collection, and completes the proof.

Colorful Helly theorem

The colorful Helly theorem is an extension of Helly's theorem in which, instead of one collection, there are ''d''+1 collections of convex subsets of . If, for ''every'' choice of a ''transversal'' – one set from every collection – there is a point in common to all the chosen sets, then for ''at least one'' of the collections, there is a point in common to all sets in the collection. Figuratively, one can consider the ''d''+1 collections to be of ''d''+1 different colors. Then the theorem says that, if every choice of one-set-per-color has a non-empty intersection, then there exists a color such that all sets of that color have a non-empty intersection.

Fractional Helly theorem

For every ''a'' > 0 there is some ''b'' > 0 such that, if are ''n'' convex subsets of , and at least an ''a''-fraction of (''d''+1)-tuples of the sets have a point in common, then a fraction of at least ''b'' of the sets have a point in common.

Notes

References