The geometry of Euclidean convolution inequalities and entropy

Authors:
Dario Cordero-Erausquin and Michel Ledoux

Journal:
Proc. Amer. Math. Soc. **138** (2010), 2755-2769

MSC (2010):
Primary 42A85, 52A40, 60E15; Secondary 60G15, 94A17

Published electronically:
April 21, 2010

Previous version:
Original version posted March 26, 2010

Corrected version:
Current version corrects publisher's introduction of $=\mu_2$ in the first sentence of the fourth paragraph and the introduction of $_n$ at the end of the second line of the fifth paragraph.

MathSciNet review:
2644890

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The goal of this paper is to show that some convolution type inequalities from Harmonic Analysis and Information Theory, such as Young's convolution inequality (with sharp constant), Nelson's hypercontractivity of the Hermite semi-group or Shannon's inequality, can be reduced to a simple geometric study of frames of . We shall derive directly entropic inequalities, which were recently proved to be dual to the Brascamp-Lieb convolution type inequalities.

**1.**Keith Ball,*Convex geometry and functional analysis*, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 161–194. MR**1863692**, 10.1016/S1874-5849(01)80006-1**2.**F. Barthe,*Optimal Young’s inequality and its converse: a simple proof*, Geom. Funct. Anal.**8**(1998), no. 2, 234–242. MR**1616143**, 10.1007/s000390050054**3.**F. Barthe, D. Cordero-Erausquin, and B. Maurey,*Entropy of spherical marginals and related inequalities*, J. Math. Pures Appl. (9)**86**(2006), no. 2, 89–99 (English, with English and French summaries). MR**2247452**, 10.1016/j.matpur.2006.04.003**4.**F. Barthe, D. Cordero-Erausquin, M. Ledoux and B. Maurey,*Correlation and Brascamp-Lieb inequalities for Markov semigroups*, preprint (2009).**5.**William Beckner,*Inequalities in Fourier analysis*, Ann. of Math. (2)**102**(1975), no. 1, 159–182. MR**0385456****6.**Jonathan Bennett and Neal Bez,*Closure properties of solutions to heat inequalities*, J. Geom. Anal.**19**(2009), no. 3, 584–600. MR**2496567**, 10.1007/s12220-009-9070-2**7.**Herm Jan Brascamp and Elliott H. Lieb,*Best constants in Young’s inequality, its converse, and its generalization to more than three functions*, Advances in Math.**20**(1976), no. 2, 151–173. MR**0412366****8.**Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao,*The Brascamp-Lieb inequalities: finiteness, structure and extremals*, Geom. Funct. Anal.**17**(2008), no. 5, 1343–1415. MR**2377493**, 10.1007/s00039-007-0619-6**9.**Eric A. Carlen,*Superadditivity of Fisher’s information and logarithmic Sobolev inequalities*, J. Funct. Anal.**101**(1991), no. 1, 194–211. MR**1132315**, 10.1016/0022-1236(91)90155-X**10.**Eric A. Carlen and Dario Cordero-Erausquin,*Subadditivity of the entropy and its relation to Brascamp-Lieb type inequalities*, Geom. Funct. Anal.**19**(2009), no. 2, 373–405. MR**2545242**, 10.1007/s00039-009-0001-y**11.**E. A. Carlen, E. H. Lieb, and M. Loss,*A sharp analog of Young’s inequality on 𝑆^{𝑁} and related entropy inequalities*, J. Geom. Anal.**14**(2004), no. 3, 487–520. MR**2077162**, 10.1007/BF02922101**12.**Amir Dembo, Thomas M. Cover, and Joy A. Thomas,*Information-theoretic inequalities*, IEEE Trans. Inform. Theory**37**(1991), no. 6, 1501–1518. MR**1134291**, 10.1109/18.104312**13.**Leonard Gross,*Logarithmic Sobolev inequalities*, Amer. J. Math.**97**(1975), no. 4, 1061–1083. MR**0420249****14.**Edward Nelson,*A quartic interaction in two dimensions*, Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965) M.I.T. Press, Cambridge, Mass., 1966, pp. 69–73. MR**0210416**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
42A85,
52A40,
60E15,
60G15,
94A17

Retrieve articles in all journals with MSC (2010): 42A85, 52A40, 60E15, 60G15, 94A17

Additional Information

**Dario Cordero-Erausquin**

Affiliation:
Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie (Paris 6), 4 place Jussieu, 75252 Paris Cedex 05, France

Email:
cordero@math.jussieu.fr

**Michel Ledoux**

Affiliation:
Institut de Mathématiques de Toulouse, Université de Toulouse, 31062 Toulouse, France

Email:
ledoux@math.univ-toulouse.fr

DOI:
https://doi.org/10.1090/S0002-9939-10-10304-9

Received by editor(s):
July 16, 2009

Published electronically:
April 21, 2010

Communicated by:
Marius Junge

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.