mathematics

Kirchberger's theorem

Kirchberger's theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Kirchberger's theorem rather than just read about it. In short: Kirchberger's theorem is a theorem in discrete geometry, on linear separability. The two-dimensional version of the theorem states that, if a finite set of red and blue points in the Euclidean plane has the property that, for every four points, there exists a line separating the red and blue points within those four, then there exists a single line separating all the red points from all the blue points.

Key takeaways

Kirchberger's theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Kirchberger's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Kirchberger's theorem from memory before moving on to harder problems.

Reference excerpt

Kirchberger's theorem is a theorem in discrete geometry, on linear separability. The two-dimensional version of the theorem states that, if a finite set of red and blue points in the Euclidean plane has the property that, for every four points, there exists a line separating the red and blue points within those four, then there exists a single line separating all the red points from all the blue points. 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. More generally, for finitely many red and blue points in d {\displaystyle d} -dimensional Euclidean space, if the red and blue points in every subset of d + 2 {\displaystyle d+2} of the points are linearly separable, then all the red points and all the blue points are linearly separable. 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 d + 2 {\displaystyle d+2} points for which the convex hulls of the red and blue points in the subsets also intersect.

History and proofs 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, and published it in 1903 in Mathematische Annalen, 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. Since Kirchberger's work, other proofs of Kirchberger's theorem have been published, including simple proofs based on Helly's theorem on intersections of convex sets, based on Carathéodory's theorem on membership in convex hulls, or based on principles related to Radon's theorem on intersections of convex hulls. However, Helly's theorem, Carathéodory's theorem, and Radon's theorem all postdate Kirchberger's theorem.

Generalizations and related results A strengthened version of Kirchberger's theorem fixes one of the given points, and only considers subsets of d + 2 {\displaystyle d+2} points that include the fixed point. If the red and blue points in each of these subsets are linearly separable, then all the red points and all the blue points are linearly separable. The theorem also holds if the red points and blue points form compact sets that are not necessarily finite. By using stereographic projection, Kirchberger's theorem can be used to prove a similar result for circular or spherical separability: if every five points of finitely many red and blue points in the plane can have their red and blue points separated by a circle, or every d + 3 {\displaystyle d+3} points in higher dimensions can have their red and blue points separated by a hypersphere, then all the red and blue points can be separated in the same way.

See also Hyperplane separation theorem, the theorem that disjoint compact convex sets are linearly separable

References

Further reading Bergold, Helena; Felsner, Stefan; Scheucher, Manfred; Schröder, Felix; Steiner, Raphael (2020), "Topological drawings meet classical theorems from convex geometry", Proceedings of the 28th International Symposium on Graph Drawing and Network Visualization, arXiv:2005.12568 Cordovil, Raul (1982), "Sur un theoreme de separation des matroïdes orientes de rang trois", Discrete Mathematics, 40 (2–3): 163–169, doi:10.1016/0012-365X(82)90117-0, MR 0676722 Houle, Michael E. (1991), "Theorems on the existence of separating surfaces", Discrete & Computational Geometry, 6 (1): 49–56, doi:10.1007/BF02574673, MR 1073072, S2CID 1992810 Lángi, Zsolt; Naszódi, Márton (2008), "Kirchberger-type theorems for separation by convex domains", Periodica Mathematica Hungarica, 57 (2): 185–196, doi:10.1007/s10998-008-8185-6, MR 2469604, S2CID 15506550 Netrebin, A. G.; Shashkin, Yu. A. (1985), "Theorems of Kirchberger and Carathéodory type in generalized convexity spaces", Doklady Akademii Nauk SSSR, 283 (5): 1085–1088, MR 0802134 Rennie, B. C. (1970), "A theorem like Kirchberger's", Journal of the London Mathematical Society, Second Series, 2: 40–44, doi:10.1112/jlms/s2-2.1.40, MR 0250192

Worked examples

Example 1 — a first encounter with Kirchberger's theorem

Start with the simplest possible case. Write down what Kirchberger's theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Kirchberger's theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Kirchberger's theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Kirchberger's theorem

In research: Kirchberger's theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Kirchberger's theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Kirchberger's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in convex geometry, Theorems in discrete geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Kirchberger's theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Kirchberger's theorem” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Kirchberger's theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Kirchberger's theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Kirchberger's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Kirchberger's theorem in simple terms?

Why does Kirchberger's theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Kirchberger's theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Kirchberger's theorem.