# Helly number

From Encyclopedia of Mathematics

$ \def\X{\mathcal X} % family of sets \def\S{\mathcal S} % subfamily $

The **Helly number** $ H(\X) $ of a family of sets $\X$ is
(in analogy to Helly's theorem) the smallest natural number $k$
such that the following (compactness-type) intersection property holds:

- Let $ \S $ be a subfamily of $ \X $. If any $k$ members of $\S$ have a common point, then the sets of $\S$ have a common point.

This is also called the *Helly property*, and
the corresponding is called a *Helly family* (of order $k$).

**How to Cite This Entry:**

Helly number.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Helly_number&oldid=30988