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 mathematicslogicstatistics 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.

Scope of Application

• 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.

• 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.

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.

Manages Complexity

Logarithmically concave measure compresses multiple mathematicslogicstatistics 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.

Abstract Reasoning

1. Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
2. 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.
3. Check operation and conditions. The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave.
4. Demand recognition evidence.

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.

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

• Projection-valued measure — 0.83
• Spherical Measure — 0.82
• Lévy–Prokhorov metric — 0.81
• Lipschitz continuity — 0.81
• Caccioppoli set — 0.81

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