Subadditivity of The Entropy and its Relation to Brascamp–Lieb Type Inequalities

@article{Carlen2007SubadditivityOT,
  title={Subadditivity of The Entropy and its Relation to Brascamp–Lieb Type Inequalities},
  author={Eric A. Carlen and Dario Cordero-Erausquin},
  journal={Geometric and Functional Analysis},
  year={2007},
  volume={19},
  pages={373-405}
}
We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on $${\mathbb {R}^n}$$, and fully… Expand

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References

SHOWING 1-10 OF 28 REFERENCES
The Brascamp–Lieb Inequalities: Finiteness, Structure and Extremals
Abstract.We consider the Brascamp–Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of theExpand
Superadditivity of Fisher's information and logarithmic Sobolev inequalities
Abstract We prove a theorem characterizing Gaussian functions and we prove a strict superaddivity property of the Fisher information. We use these results to determine the cases of equality in theExpand
Best Constants in Young's Inequality, Its Converse, and Its Generalization to More than Three Functions
The best possible constant Dmt in the inequality | ∬ dx dyf(x)g(x —y) h(y)| |, 1/p + llq+ 1/t = 2, is determined; the equality is reached if /, g, and h are appropriate Gaussians. The same is shownExpand
A sharp analog of Young’s inequality on SN and related entropy inequalities
We prove a sharp analog of Young’s inequality on SN, and deduce from it certain sharp entropy inequalities. The proof turns on constructing a nonlinear heat flow that drives trial functions toExpand
Chapter 4 – Convex Geometry and Functional Analysis
This chapter describes three topics that lie at the intersection of functional analysis, harmonic analysis, probability theory, and convex geometry. The classical isoperimetric inequality states thatExpand
On a reverse form of the Brascamp-Lieb inequality
Abstract. We prove a reverse form of the multidimensional Brascamp-Lieb inequality. Our method also gives a new way to derive the Brascamp-Lieb inequality and is rather convenient for the study ofExpand
Optimisers for the Brascamp-Lieb inequality
We find all optimisers for the Brascamp-Lieb inequality, thus completing the problem which was settled in special cases by Barthe; Carlen, Lieb and Loss; and Bennett, Carbery, Christ and Tao. OurExpand
Entropy of spherical marginals and related inequalities
We investigate relations between the entropy of a spherical density and those of its marginals, together with spherical convolution type inequalities. We extend results by Carlen, Lieb and Loss toExpand
Finite bounds for Hölder-Brascamp-Lieb multilinear inequalities
A criterion is established for the validity of multilinear inequalities of a class considered by Brascamp and Lieb, generalizing well-known inequalties of Rogers and H\"older, Young, andExpand
Inverse Brascamp-Lieb Inequalities along the Heat Equation
Adapting Borell’s proof of Ehrhard’s inequality for general sets, we provide a semi-group approach to the reverse Brascamp-Lieb inequality, in its “convexity” version.
...
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