mathematics

Stromquist–Woodall theorem

Stromquist–Woodall 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 Stromquist–Woodall theorem rather than just read about it. In short: The Stromquist–Woodall theorem is a theorem in fair division and measure theory. Informally, it says that, for any cake, for any n people with different tastes, and for any fraction w, there exists a subset of the cake that all people value at exactly a fraction w of the total cake value, and it can be cut using at most 2 n − 2 {\displaystyle 2n-2} cuts.

Key takeaways

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

Reference excerpt

The Stromquist–Woodall theorem is a theorem in fair division and measure theory. Informally, it says that, for any cake, for any n people with different tastes, and for any fraction w, there exists a subset of the cake that all people value at exactly a fraction w of the total cake value, and it can be cut using at most 2 n − 2 {\displaystyle 2n-2} cuts. The theorem is about a circular 1-dimensional cake (a "pie"). Formally, it can be described as the interval [0,1] in which the two endpoints are identified. There are n continuous measures over the cake: V 1 , … , V n {\displaystyle V_{1},\ldots ,V_{n}} ; each measure represents the valuations of a different person over subsets of the cake. The theorem says that, for every weight w ∈ [ 0 , 1 ] {\displaystyle w\in [0,1]} , there is a subset C w {\displaystyle C_{w}} , which all people value at exactly w {\displaystyle w} :

∀ i = 1 , … , n : V i ( C w ) = w {\displaystyle \forall i=1,\ldots ,n:\,\,\,\,\,V_{i}(C_{w})=w} , where C w {\displaystyle C_{w}} is a union of at most n − 1 {\displaystyle n-1} intervals. This means that 2 n − 2 {\displaystyle 2n-2} cuts are sufficient for cutting the subset C w {\displaystyle C_{w}} . If the cake is not circular (that is, the endpoints are not identified), then C w {\displaystyle C_{w}} may be the union of up to n {\displaystyle n} intervals, in case one interval is adjacent to 0 and one other interval is adjacent to 1.

Proof sketch Let W ⊆ [ 0 , 1 ] {\displaystyle W\subseteq [0,1]} be the subset of all weights for which the theorem is true. Then:

1 ∈ W {\displaystyle 1\in W} . Proof: take C 1 := C {\displaystyle C_{1}:=C} (recall that the value measures are normalized such that all partners value the entire cake as 1). If w ∈ W {\displaystyle w\in W} , then also 1 − w ∈ W {\displaystyle 1-w\in W} . Proof: take C 1 − w := C ∖ C w {\displaystyle C_{1-w}:=C\smallsetminus C_{w}} . If C w {\displaystyle C_{w}} is a union of n − 1 {\displaystyle n-1} intervals in a circle, then C 1 − w {\displaystyle C_{1-w}} is also a union of n − 1 {\displaystyle n-1} intervals.

W {\displaystyle W} is a closed set. This is easy to prove, since the space of unions of n − 1 {\displaystyle n-1} intervals is a compact set under a suitable topology. If w ∈ W {\displaystyle w\in W} , then also w / 2 ∈ W {\displaystyle w/2\in W} . This is the most interesting part of the proof; see below. From 1-4, it follows that W = [ 0 , 1 ] {\displaystyle W=[0,1]} . In other words, the theorem is valid for every possible weight.

Proof sketch for part 4 Assume that C w {\displaystyle C_{w}} is a union of n − 1 {\displaystyle n-1} intervals and that all n {\displaystyle n} partners value it as exactly w {\displaystyle w} . Define the following function on the cake, f : C → R n {\displaystyle f:C\to \mathbb {R} ^{n}} :

f ( t ) = ( t , t 2 , … , t n ) t ∈ [ 0 , 1 ] {\displaystyle f(t)=(t,t^{2},\ldots ,t^{n})\,\,\,\,\,\,t\in [0,1]}

Define the following measures on R n {\displaystyle \mathbb {R} ^{n}} :

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Stromquist–Woodall theorem

Start with the simplest possible case. Write down what Stromquist–Woodall 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 Stromquist–Woodall 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 Stromquist–Woodall 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 Stromquist–Woodall theorem

In research: Stromquist–Woodall 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 Stromquist–Woodall 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: Stromquist–Woodall theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Cake-cutting, Theorems in measure theory, so understanding it makes those chapters shorter.
In everyday life: Look for Stromquist–Woodall 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 Stromquist–Woodall theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Stromquist–Woodall 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 Stromquist–Woodall theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Stromquist–Woodall theorem in simple terms?

Why does Stromquist–Woodall 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 Stromquist–Woodall 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 Stromquist–Woodall theorem.