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Helly’s Theorem

The following important basic combinatorial result on convex sets was proved by Eduard Helly in 1913. Giving his own proof Radon published the result earlier in 1921. Helly published his proof  only in 1923 [1] .  Radon’s proof is based on Radon’s theorem [2] (for historical details see [3] ). In 1930 he published a generalization of the result  [4] . An elementary proof of Helly’s theorem can be found in [5] . Here it is also shown that Carathéodory’s theorem on convex hulls may be derived from Helly’s theorem thus showing its central role in the theory of convex bodies.

Helly’s Theorem: Let typeset structure, typeset structure, be convex sets in typeset structure. If every typeset structure of these sets have a point in common then there is a point common to all typeset structure.

Equivalently: If typeset structure, then there exists typeset structure sets typeset structure such that typeset structure and typeset structure.

Example: If in a finite system of closed intervals on a line every two of them have a point in common, then the intersection of the all intervals is non-empty.

Example: Given three convex sets in typeset structure having a point in common, then Helly’s theorem shows that any convex set containing points of intersections of any two of them contains a point from their common intersection.

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The theorem is also true for infinite collections if one assumes that the sets are compact. Without an additional condition the original Helly’s theorem is not true for infinite collections. Consider for instance the system of all upper half-planes 1. Then any three of them have a non-empty intersection, but all of them have no common point.

The number typeset structure cannot be replaced by typeset structure in general. However under some additional conditions it can be reduced. For instance, if in the plane we have a system of rectangles with parallel sides such that every two of them have a common point then there is a point common to all  of them.

Helly’s theorem gave rise to the notion of Helly family and Helly number.

Notes

1 An upper half-plane is a planar region consisting of all points on one side of an infinite straight line parallel to the typeset structure-axis, and no points on the other side.

References

[1]  Helly, E. (1923). Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jber. Deutsch. Math. Verein., 32, 175-176.

[2]  Radon, J. (1921). Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Math. Ann., 83, 113-115.

[3]  Danzer, L., Grünbaum, B., & Klee, V. (1963). Helly’s Theorem and its Relatives. Providence: Amer. Math. Soc..

[4]  Helly, E. (1930). Über Systeme von agbeschlossenen Mengen mit gemeinschaflichen Punkten. Monatsh. Math. Verein, 37, 281-302.

[5]  Rabin, M. (1955). A note on Helly’s theorem. Pacific J. Math., 3, 363-366.

Cite this web-page as:

Štefan Porubský: Helly’s Theorem.

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