mathematics

Integrally convex set

Integrally convex set 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 Integrally convex set rather than just read about it. In short: An integrally convex set is the discrete geometry analogue of the concept of convex set in geometry. A subset X of the integer grid Z n {\displaystyle \mathbb {Z} ^{n}} is integrally convex if any point y in the convex hull of X can be expressed as a convex combination of the points of X that are "near" y, where "near" means that the distance between each two coordinates is less than 1.

Key takeaways

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

Reference excerpt

An integrally convex set is the discrete geometry analogue of the concept of convex set in geometry. A subset X of the integer grid Z n {\displaystyle \mathbb {Z} ^{n}} is integrally convex if any point y in the convex hull of X can be expressed as a convex combination of the points of X that are "near" y, where "near" means that the distance between each two coordinates is less than 1.

Definitions Let X be a subset of Z n {\displaystyle \mathbb {Z} ^{n}} . Denote by ch(X) the convex hull of X. Note that ch(X) is a subset of R n {\displaystyle \mathbb {R} ^{n}} , since it contains all the real points that are convex combinations of the integer points in X. For any point y in R n {\displaystyle \mathbb {R} ^{n}} , denote near(y) := {z in Z n {\displaystyle \mathbb {Z} ^{n}} | |zi - yi| < 1 for all i in {1,...,n} }. These are the integer points that are considered "nearby" to the real point y. A subset X of Z n {\displaystyle \mathbb {Z} ^{n}} is called integrally convex if every point y in ch(X) is also in ch(X ∩ near(y)).

Example

Let n = 2 and let X = { (0,0), (1,0), (2,0), (2,1) }. Its convex hull ch(X) contains, for example, the point y = (1.2, 0.5). The integer points nearby y are near(y) = {(1,0), (2,0), (1,1), (2,1) }. So X ∩ near(y) = {(1,0), (2,0), (2,1)}. But y is not in ch(X ∩ near(y)). See image at the right. Therefore X is not integrally convex. In contrast, the set Y = { (0,0), (1,0), (2,0), (1,1), (2,1) } is integrally convex.

Properties Iimura, Murota and Tamura have shown the following property of integrally convex set. Let X ⊂ Z n {\displaystyle X\subset \mathbb {Z} ^{n}} be a finite integrally convex set. There exists a triangulation of ch(X) that is integral, i.e.:

The vertices of the triangulation are the vertices of X; The vertices of every simplex of the triangulation lie in the same "cell" (hypercube of side-length 1) of the integer grid Z n {\displaystyle \mathbb {Z} ^{n}} .

The example set X is not integrally convex, and indeed ch(X) does not admit an integral triangulation: every triangulation of ch(X), either has to add vertices not in X, or has to include simplices that are not contained in a single cell. In contrast, the set Y = { (0,0), (1,0), (2,0), (1,1), (2,1) } is integrally convex, and indeed admits an integral triangulation, e.g. with the three simplices {(0,0),(1,0),(1,1)} and {(1,0),(2,0),(2,1)} and {(1,0),(1,1),(2,1)}. See image at the right.

References

Illustrations

Worked examples

Example 1 — a first encounter with Integrally convex set

Start with the simplest possible case. Write down what Integrally convex set 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 Integrally convex set 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 Integrally convex set 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 Integrally convex set

In research: Integrally convex set 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 Integrally convex set 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: Integrally convex set is common in secondary-school and first-year university syllabi. It links to neighbouring topics Discrete geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Integrally convex set 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 Integrally convex set in 20 minutes

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

Frequently asked questions

What is Integrally convex set in simple terms?

Why does Integrally convex set 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 Integrally convex set?

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 Integrally convex set.