Logarithmically concave measure¶

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

Version: v1 · 2026-09-28 · History
Domain-specific #: 10476
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Convex Geometry, Measure Theory → Mathematics

Core Idea¶

Logarithmically concave measure is treated here as the recurring mathematics_logic_statistics identity summarized by this source-grounded definition: In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has.

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

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. The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave.

For Logarithmically concave measure, the abstraction is narrower than the article's general subject matter: a positive case must preserve In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — The restriction of the Lebesgue measure to any convex set is also log-concave.
Constitutive relation — 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.
Operating condition — The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave.
Recognition evidence — The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave.
Admissible variation — In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has.
Characteristic consequence — \mu(\lambda A + (1-\lambda) B) \geq \mu(A)^\lambda \mu(B)^{1-\lambda},.
Failure boundary — where λ A + (1 − λ) B denotes the Minkowski sum of λ A and (1 − λ) B.

What It Is Not¶

Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has.
Not an over-broad reading. The restriction of the Lebesgue measure to any convex set is also log-concave.
Not an over-broad reading. 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.
Not an over-broad reading. The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave.
Not automatically Log-Sum Inequality. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Logarithmically concave measure applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Examples. 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.
Examples. The restriction of the Lebesgue measure to any convex set is also log-concave.
Examples. The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave.
Examples. The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave.
Documented setting. In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has.
Documented setting. \mu(\lambda A + (1-\lambda) B) \geq \mu(A)^\lambda \mu(B)^{1-\lambda},.

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Logarithmically concave measure names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has. The strongest recognition evidence in the frozen account is: The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification The restriction of the Lebesgue measure to any convex set is also log-concave. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Logarithmically concave measure compresses multiple mathematics_logic_statistics details into a stable diagnostic relation. The source shows both the central mechanism—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.—and the practical consequence—\mu(\lambda A + (1-\lambda) B) \geq \mu(A)^\lambda \mu(B)^{1-\lambda},. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics_logic_statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has.
Check operation and conditions. The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave.
Demand recognition evidence. The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave.
Test variation. Change an implementation or setting while preserving in mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Logarithmically concave measure transfers literally when a new case preserves the same carrier type, relation, and recognition test. 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. The restriction of the Lebesgue measure to any convex set is also log-concave.

Beyond the home domain. No canonical parent is asserted for Logarithmically concave measure. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

The restriction of the Lebesgue measure to any convex set is also log-concave. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has; recognition evidence → The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave

Applied / In Practice¶

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. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Examples; invariant → In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has; boundary → the case exits the class when the restriction of the Lebesgue measure to any convex set is also log-concave

Structural Tensions¶

T1 — Stable identity versus admissible variation. The restriction of the Lebesgue measure to any convex set is also log-concave. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. 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. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. The restriction of the Lebesgue measure to any convex set is also log-concave. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Logarithmically concave measure literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. 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. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Logarithmically concave measure distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Logarithmically concave measure is structural-leaning. Its structural side is the repeatable organization summarized by In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has. Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: 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. It further constrains recognition and variation through: The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave. The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave.

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Logarithmically concave measure literal. Its documented scope includes the condition that 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. Another bounded application condition is that The restriction of the Lebesgue measure to any convex set is also log-concave. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has.—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Logarithmically concave measure. The reviewed identity is: In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

Logarithmically concave measure sits in a sparse region of the domain-specific corpus (85^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has?
Log-Sum Inequality. A convexity inequality showing that the weighted sum of local log ratios between paired nonnegative masses is at least the corresponding log ratio after aggregation, with equality exactly under proportional local ratios. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Null Set. Classify a measurable subset as negligible when its measure is zero, allowing it to be ignored by almost-everywhere statements without requiring it to be empty. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Binary entropy function. The Shannon entropy of a Bernoulli variable as a concave function of its success probability. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Logarithmically concave measure remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Logarithmically_concave_measure (revision 1133688715).

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.