mathematics

Reach (mathematics)

Reach (mathematics) 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 Reach (mathematics) rather than just read about it. In short: Let X be a subset of Rn. Then the reach of X is defined as reach ( X ) := sup { r ∈ R : ∀ x ∈ R n ∖ X with d i s t ( x , X ) < r exists a unique closest point y ∈ X such that d i s t ( x , y ) = d i s t ( x , X ) } . {\displaystyle {\text{reach}}(X):=\sup\{r\in \mathbb {R} :\forall x\in \mathbb {R} ^{n}\setminus X{\text{ with }}{\rm {dist}}(x,X)<r{\text{ exists a unique closest point }}y\in X{\text{ such that }}{\rm…

Key takeaways

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

Reference excerpt

Let X be a subset of Rn. Then the reach of X is defined as

reach ( X ) := sup { r ∈ R : ∀ x ∈ R n ∖ X with d i s t ( x , X ) < r exists a unique closest point y ∈ X such that d i s t ( x , y ) = d i s t ( x , X ) } . {\displaystyle {\text{reach}}(X):=\sup\{r\in \mathbb {R} :\forall x\in \mathbb {R} ^{n}\setminus X{\text{ with }}{\rm {dist}}(x,X)<r{\text{ exists a unique closest point }}y\in X{\text{ such that }}{\rm {dist}}(x,y)={\rm {dist}}(x,X)\}.}

Examples Shapes that have reach infinity include

a single point, a straight line, a full square, and any convex set. The graph of ƒ(x) = |x| has reach zero. A circle of radius r has reach r.

References Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, vol. 153, New York: Springer-Verlag New York Inc., pp. xiv+676, ISBN 978-3-540-60656-7, MR 0257325, Zbl 0176.00801

Worked examples

Example 1 — a first encounter with Reach (mathematics)

Start with the simplest possible case. Write down what Reach (mathematics) 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 Reach (mathematics) 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 Reach (mathematics) 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 Reach (mathematics)

In research: Reach (mathematics) 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 Reach (mathematics) 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: Reach (mathematics) is common in secondary-school and first-year university syllabi. It links to neighbouring topics Geometric measurement, Real analysis, Topology, so understanding it makes those chapters shorter.
In everyday life: Look for Reach (mathematics) 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 Reach (mathematics) in 20 minutes

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

Frequently asked questions

What is Reach (mathematics) in simple terms?

Let X be a subset of Rn. Then the reach of X is defined as reach ( X ) := sup { r ∈ R : ∀ x ∈ R n ∖ X with d i s t ( x , X ) < r exists a unique closest point y ∈ X such that d i s t ( x , y ) = d i s t ( x , X ) } . {\displaystyle {\text{reach}}(X):=\sup\{r\in \mathbb {R} :\forall x\in \mathbb {R}…

Why does Reach (mathematics) 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 Reach (mathematics)?

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 Reach (mathematics).