The area measure of log-concave functions and related inequalities

Andrea Colesanti; Ilaria Fragala'

RECORD

The area measure of log-concave functions and related inequalities

Andrea Colesanti, Ilaria Fragala'

Functional Analysis · Submitted 12 Dec 2011

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In the same framework, we obtain a functional counterpart of Minkowski first inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗

Cite ▾

Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,753 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.69 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.16 · compares with prior results 0.36 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

On the class of log-concave functions on $\R^n$, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions. We prove some integral representation formulae for such first variation, which lead to define in a natural way the notion of area measure for a log-concave function. In the same framework, we obtain a functional counterpart of Minkowski first inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures. Finally, we propose a suitable functional version of the classical Minkowski problem for convex bodies, and prove some partial results towards its solution.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	On the class of log-concave functions on $\R^n$, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions.	contribution 0.95	0.91	0.07	asserted 0.97
s1	We prove some integral representation formulae for such first variation, which lead to define in a natural way the notion of area measure for a log-concave function.	contribution 0.77	0.96	0.16	asserted 1.00
s2	In the same framework, we obtain a functional counterpart of Minkowski first inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures.	result 0.97	0.96	0.48	asserted 1.00
s3	Finally, we propose a suitable functional version of the classical Minkowski problem for convex bodies, and prove some partial results towards its solution.	contribution 0.85	0.94	0.16	asserted 0.86

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

On the class of log-concave functions on $\R^n$, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions. We prove some integral representation formulae for such first variation, which lead to define in a natural way the notion of area measure for a log-concave function. In the same framework, we obtain a functional counterpart of Minkowski first inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures. Finally, we propose a suitable functional version of the classical Minkowski problem for convex bodies, and prove some partial results towards its solution.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

2 arXiv papers cite this in 6 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 5 sentences without a confident label

Uses its method or tool 1 paper

Inequality (6.2), together with the corresponding equality case, can be obtained by applying Theorem 5.1 in [ref] with $g = χ_B$.

Quermassintegrals of quasi-concave functions and generalized Pr\'ekopa-Leindler inequalities · Functional inequalities · 0.81

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.