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Radon's theorem

Radon'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 Radon's theorem rather than just read about it. In short: In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that:Any set of d + 2 points in Rd can be partitioned into two sets whose convex hulls intersect. A point in the intersection of these convex hulls is called a Radon point of the set.For example, in the case d = 2, any set of four points in the Euclidean plane can be partitioned in one of two ways.

Radon's theorem — main illustration
Radon's theorem — illustration

Key takeaways

  • Radon'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 Radon's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Radon's theorem from memory before moving on to harder problems.

Reference excerpt

In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that:Any set of d + 2 points in Rd can be partitioned into two sets whose convex hulls intersect. A point in the intersection of these convex hulls is called a Radon point of the set.For example, in the case d = 2, any set of four points in the Euclidean plane can be partitioned in one of two ways. It may form a triple and a singleton, where the convex hull of the triple (a triangle) contains the singleton; alternatively, it may form two pairs of points that form the endpoints of two intersecting line segments.

Proof and construction Consider any set X = { x 1 , x 2 , … , x d + 2 } ⊂ R d {\displaystyle X=\{x_{1},x_{2},\dots ,x_{d+2}\}\subset \mathbf {R} ^{d}} of d + 2 points in d-dimensional space. Then there exists a set of multipliers a1, ..., ad + 2, not all of which are zero, solving the system of linear equations

∑ i = 1 d + 2 a i x i = 0 , ∑ i = 1 d + 2 a i = 0 , {\displaystyle \sum _{i=1}^{d+2}a_{i}x_{i}=0,\quad \sum _{i=1}^{d+2}a_{i}=0,}

because there are d + 2 unknowns (the multipliers) but only d + 1 equations that they must satisfy (one for each coordinate of the points, together with a final equation requiring the sum of the multipliers to be zero). Fix some particular nonzero solution a1, ..., ad + 2. Let I ⊆ X {\displaystyle I\subseteq X} be the set of points with positive multipliers, and let J = X ∖ I {\displaystyle J=X\setminus I} be the set of points with multipliers that are negative or zero. Then I {\displaystyle I} and J {\displaystyle J} form the required partition of the points into two subsets with intersecting convex hulls. The convex hulls of I {\displaystyle I} and J {\displaystyle J} must intersect, because they both contain the point

p = ∑ x i ∈ I a i A x i = ∑ x j ∈ J − a j A x j , {\displaystyle p=\sum _{x_{i}\in I}{\frac {a_{i}}{A}}x_{i}=\sum _{x_{j}\in J}{\frac {-a_{j}}{A}}x_{j},}

where

A = ∑ x i ∈ I a i = − ∑ x j ∈ J a j . {\displaystyle A=\sum _{x_{i}\in I}a_{i}=-\sum _{x_{j}\in J}a_{j}.}

The left hand side of the formula for p {\displaystyle p} expresses this point as a convex combination of the points in I {\displaystyle I} , and the right hand side expresses it as a convex combination of the points in J {\displaystyle J} . Therefore, p {\displaystyle p} belongs to both convex hulls, completing the proof. This proof method allows for the efficient construction of a Radon point, in an amount of time that is polynomial in the dimension, by using Gaussian elimination or other efficient algorithms to solve the system of equations for the multipliers.

… excerpt ends here. Continue reading the full article.

Illustrations

Radon's theorem: Two sets of four points in the plane (the vertices of a square and an equilateral triangle with its centroid), the multipliers solving the system of three linear equations for these points, and the Radon partitions formed by separating the points with positive multipliers from the points with negative multipliers.
Two sets of four points in the plane (the vertices of a square and an equilateral triangle with its centroid), the multipliers solving the system of three linear equations for these points, and the Radon partitions formed by separating the points with positive multipliers from the points with negative multipliers.

Worked examples

Example 1 — a first encounter with Radon's theorem

Start with the simplest possible case. Write down what Radon'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 Radon'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 Radon'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 Radon's theorem

In research
Radon'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 Radon'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
Radon's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Convex hulls, Geometric transversal theory, Theorems in convex geometry, so understanding it makes those chapters shorter.
In everyday life
Look for Radon'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.
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How to study Radon's theorem in 20 minutes

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

Frequently asked questions

What is Radon's theorem in simple terms?

In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that:Any set of d + 2 points in Rd can be partitioned into two sets whose convex hulls intersect. A point in the intersection of these convex hulls is called a Radon point of the set.For example, in the case d =…

Why does Radon'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 Radon'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 Radon's theorem.

Tags

  • Convex hulls
  • Geometric transversal theory
  • Theorems in convex geometry
  • Theorems in discrete geometry

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