Donald Watson phrases this result more colorfully, with a farmyard analogy: If sheep and goats are grazing in a field and for every four animals there exists a line separating the sheep from the goats then there exists such a line for all the animals.
Another equivalent way of stating the result is that, if the convex hulls of finitely many red and blue points have a nonempty intersection, then there exists a subset of
[2][3] The theorem is named after German mathematician Paul Kirchberger, a student of David Hilbert at the University of Göttingen who proved it in his 1902 dissertation,[4] and published it in 1903 in Mathematische Annalen,[5] as an auxiliary theorem used in his analysis of Chebyshev approximation.
A report of Hilbert on the dissertation states that some of Kirchberger's auxiliary theorems in this part of his dissertation were known to Hermann Minkowski but unpublished; it is not clear whether this statement applies to the result now known as Kirchberger's theorem.
A strengthened version of Kirchberger's theorem fixes one of the given points, and only considers subsets of