science

Kovner–Besicovitch measure

Kovner–Besicovitch measure is a science 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 Kovner–Besicovitch measure rather than just read about it. In short: In plane geometry the Kovner–Besicovitch measure is a number defined for any bounded convex set describing how close to being centrally symmetric it is. It is the fraction of the area of the set that can be covered by its largest centrally symmetric subset.

Key takeaways

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

Reference excerpt

In plane geometry the Kovner–Besicovitch measure is a number defined for any bounded convex set describing how close to being centrally symmetric it is. It is the fraction of the area of the set that can be covered by its largest centrally symmetric subset.

Properties This measure is one for a set that is centrally symmetric, and less than one for sets whose closure is not centrally symmetric. It is invariant under affine transformations of the plane. If c {\displaystyle c} is the center of symmetry of the largest centrally-symmetric set within a given convex body K {\displaystyle K} , then the centrally-symmetric set itself is the intersection of K {\displaystyle K} with its reflection across c {\displaystyle c} .

Minimizers The convex sets with the smallest possible Kovner–Besicovitch measure are the triangles, for which the measure is 2/3. The result that triangles are the minimizers of this measure is known as Kovner's theorem or the Kovner–Besicovitch theorem, and the inequality bounding the measure above 2/3 for all convex sets is the Kovner–Besicovitch inequality. The curve of constant width with the smallest possible Kovner–Besicovitch measure is the Reuleaux triangle.

Computational complexity The Kovner–Besicovitch measure of any given convex polygon with n {\displaystyle n} vertices can be found in time O ( n log ⁡ n ) {\displaystyle O(n\log n)} by determining a translation of the reflection of the polygon that has the largest possible overlap with the unreflected polygon.

History Branko Grünbaum writes that the Kovner–Besicovitch theorem was first published in Russian, in a 1935 textbook on the calculus of variations by Mikhail Lavrentyev and Lazar Lyusternik, where it was credited to Soviet mathematician and geophysicist S. S. Kovner. Additional proofs were given by Abram Samoilovitch Besicovitch and by István Fáry, who also proved that every minimizer of the Kovner–Besicovitch measure is a triangle.

See also Estermann measure, a measure of central symmetry defined using supersets in place of subsets

References

External links A Measure of Central Symmetry, Tanya Khovanova's Math Blog, September 2, 2012

Illustrations

Worked examples

Example 1 — a first encounter with Kovner–Besicovitch measure

Start with the simplest possible case. Write down what Kovner–Besicovitch measure claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Kovner–Besicovitch measure 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 Kovner–Besicovitch measure 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 Kovner–Besicovitch measure

In research: Kovner–Besicovitch measure appears in science 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 Kovner–Besicovitch measure 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: Kovner–Besicovitch measure is common in secondary-school and first-year university syllabi. It links to neighbouring topics Euclidean symmetries, so understanding it makes those chapters shorter.
In everyday life: Look for Kovner–Besicovitch measure 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 Kovner–Besicovitch measure in 20 minutes

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

Frequently asked questions

What is Kovner–Besicovitch measure in simple terms?

Why does Kovner–Besicovitch measure matter?

Because it connects several science 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 Kovner–Besicovitch measure?

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 Kovner–Besicovitch measure.