Abstract
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 determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of the classical Brascamp–Lieb inequality, and also a fully explicit determination of all of the cases of equality. We also deduce several other consequences of the general subadditivity inequality, including a generalization of Hadamard’s inequality for determinants. Finally, we also prove a second duality theorem relating superadditivity of the Fisher information and a sharp convolution type inequality for the fundamental eigenvalues of Schrödinger operators. Though we focus mainly on the case of random variables in \({\mathbb {R}^n}\) in this paper, we discuss extensions to other settings as well.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. Ball, Convex geometry and functional analysis, in “Handbook of the Geometry of Banach Spaces, Vol. I, (W. Johnson, J. Lindenstrauss, eds.), North-Holland, Amsterdam (2001), 161–194.
Barthe F.: On a reverse form of the Brascamp–Lieb inequality. Invent. Math. 134(2), 235–361 (1998)
F. Barthe, D. Cordero–Erausquin, Inverse Brascamp–Lieb inequalities along the heat equation, Geometric Aspects of Functional Analysis (2002–2003) (V. Milman, G. Schechtman, eds.), Springer Lecture Notes in Mathematics 1850 (2004), 65–71.
Barthe F., Cordero–Erausquin D., Maurey B.: Entropy of spherical marginals and related inequalities. J. Math. Pures Appl. 86(2), 89–99 (2006)
F. Barthe, D. Cordero–Erausquin, M. Ledoux, B. Maurey, work in progress).
Bennett J., Carbery A., Christ M., Tao T.: The Brascamp–Lieb inequalities: finiteness, structure, and extremals, GAFA. Geom. funct. anal. 17(5), 1343–1415 (2007)
J. Bennett, A. Carbery, M. Christ, T. Tao, Finite bounds for Holder– Brascamp–Lieb multilinear inequalities, Math. Res. Lett, to appear.
Borell C.: Diffusion equations and geometric inequalities. Potential Anal. 12(1), 49–71 (2000)
Brascamp H., Lieb E.: The best constant in Young’s inequality and its generalization to more than three functions. Advances in Math. 20(2), 151–173 (1976)
Carlen E.: Superadditivity of Fisher information and logarithmic Sobolev inequalities. J. Funct. Anal. 101, 194–211 (1991)
Carlen E., Loss M., Lieb E.: A sharp form of Young’s inequality on S N and related entropy inequalities. Jour. Geom. Anal. 14, 487–520 (2004)
Carlen E., Loss M., Lieb E.: A inequality of Hadamard type for permanents. Meth. and Appl. of Anal. 13(1), 1–17 (2006)
Valdimarsson S.I.: Optimisers for the Brascamp–Lieb inequality. Israel J. Math. 168, 253–274 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
E.A.C’s work is partially supported by US National Science Foundation grant DMS 06-00037.
Rights and permissions
About this article
Cite this article
Carlen, E.A., Cordero–Erausquin, D. Subadditivity of The Entropy and its Relation to Brascamp–Lieb Type Inequalities. Geom. Funct. Anal. 19, 373–405 (2009). https://doi.org/10.1007/s00039-009-0001-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-009-0001-y