Helly's theorem

File:Helly's theorem.svg

Helly's theorem for the Euclidean plane: if a family of convex sets has a nonempty intersection for every triple of sets, then the whole family has a nonempty intersection.

Helly's theorem is a basic result in discrete geometry on the intersection of convex sets. It was discovered by Eduard Helly in 1913,<ref>Template:Harvtxt.</ref> but not published by him until 1923, by which time alternative proofs by Template:Harvtxt and Template:Harvtxt had already appeared. Helly's theorem gave rise to the notion of a Helly family.

Statement

Let Template:Math be a finite collection of convex subsets of <math>\R^d</math>, with <math>n\geq d+1</math>. If the intersection of every <math>d+1</math> of these sets is nonempty, then the whole collection has a nonempty intersection; that is,

<math>\bigcap_{j=1}^n X_j\ne\varnothing.</math>

For infinite collections one has to assume compactness:

Let <math>\{X_\alpha\}</math> be a collection of compact convex subsets of <math>\R^d</math>, such that every subcollection of cardinality at most <math>d+1</math> 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 Template:Harvtxt. The infinite version then follows by the finite intersection property 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 Template:Math. By our assumptions, for every Template:Math there is a point Template:Math that is in the common intersection of all Template:Math with the possible exception of Template:Math. Now we apply Radon's theorem to the set Template:Math which furnishes us with disjoint subsets Template:Math of Template:Mvar such that the convex hull of Template:Math intersects the convex hull of Template:Math. Suppose that Template:Mvar is a point in the intersection of these two convex hulls. We claim that

<math>p\in\bigcap_{j=1}^n X_j.</math>

Indeed, consider any Template:Math We shall prove that Template:Math Note that the only element of Template:Mvar that may not be in Template:Math is Template:Math. If Template:Math, then Template:Math, and therefore Template:Math. Since Template:Math is convex, it then also contains the convex hull of Template:Math and therefore also Template:Math. Likewise, if Template:Math, then Template:Math, and by the same reasoning Template:Math. Since Template:Mvar is in every Template:Math, it must also be in the intersection.

Above, we have assumed that the points Template:Math are all distinct. If this is not the case, say Template:Math for some Template:Math, then Template:Math is in every one of the sets Template:Math, and again we conclude that the intersection is nonempty. This completes the proof in the case Template:Math.

Inductive Step: Suppose Template:Math and that the statement is true for Template:Math. The argument above shows that any subcollection of Template:Math sets will have nonempty intersection. We may then consider the collection where we replace the two sets Template:Math and Template:Math with the single set Template:Math. In this new collection, every subcollection of Template:Math 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.

Template:AnchorColorful 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 Template:Math.

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.<ref name=":0">Template:Citation</ref>

Template:AnchorFractional Helly theorem

For every a > 0 there is some b > 0 such that, if Template:Math are n convex subsets of Template:Math, 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.<ref name=":0" />

See also

• Carathéodory's theorem
• Doignon's theorem
• Kirchberger's theorem
• Shapley–Folkman lemma
• Krein–Milman theorem
• Choquet theory
• Radon's theorem, and its generalization, Tverberg's theorem
• Cantor's intersection theorem - another theorem on intersection of sets
• Helly Family

References

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• Heinrich Guggenheimer (1977) Applicable Geometry, page 137, Krieger, Huntington Template:ISBN .
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