science

Logarithmically concave measure

Logarithmically concave 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 Logarithmically concave measure rather than just read about it. In short: In mathematics, a Borel measure μ on n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of R n {\displaystyle \mathbb {R} ^{n}} and 0 < λ < 1, one has μ ( λ A + ( 1 − λ ) B ) ≥ μ ( A ) λ μ ( B ) 1 − λ , {\displaystyle \mu (\lambda A+(1-\lambda )B)\geq \mu (A)^{\lambda }\mu (B)^{1-\lambda },} where λ A + (…

Key takeaways

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

Reference excerpt

In mathematics, a Borel measure μ on n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of R n {\displaystyle \mathbb {R} ^{n}} and 0 < λ < 1, one has

μ ( λ A + ( 1 − λ ) B ) ≥ μ ( A ) λ μ ( B ) 1 − λ , {\displaystyle \mu (\lambda A+(1-\lambda )B)\geq \mu (A)^{\lambda }\mu (B)^{1-\lambda },}

where λ A + (1 − λ) B denotes the Minkowski sum of λ A and (1 − λ) B.

Examples The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave. The restriction of the Lebesgue measure to any convex set is also log-concave. By a theorem of Borell, a probability measure on R^d is log-concave if and only if it has a density with respect to the Lebesgue measure on some affine hyperplane, and this density is a logarithmically concave function. Thus, any Gaussian measure is log-concave. The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave.

See also Convex measure, a generalisation of this concept Logarithmically concave function

References

Worked examples

Example 1 — a first encounter with Logarithmically concave measure

Start with the simplest possible case. Write down what Logarithmically concave 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 Logarithmically concave 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 Logarithmically concave 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 Logarithmically concave measure

In research: Logarithmically concave 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 Logarithmically concave 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: Logarithmically concave measure is common in secondary-school and first-year university syllabi. It links to neighbouring topics Measures (measure theory), so understanding it makes those chapters shorter.
In everyday life: Look for Logarithmically concave 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 Logarithmically concave measure in 20 minutes

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

Frequently asked questions

What is Logarithmically concave measure in simple terms?

Why does Logarithmically concave 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 Logarithmically concave 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 Logarithmically concave measure.