Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T19:18:35.159Z Has data issue: false hasContentIssue false
  • English
  • Français

The Brascamp–Lieb Polyhedron

Published online by Cambridge University Press:  20 November 2018

Stefán Ingi Valdimarsson
Stefán Ingi Valdimarsson*
Affiliation:
UCLA Mathematics Department, Los Angeles, CA, U.S.A. and Science Institute, University of Iceland, Reykjavik, Iceland, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A set of necessary and sufficient conditions for the Brascamp–Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank 1. This complements the result of Barthe concerning the case where the linear maps all have rank 1. Pushing our analysis further, we describe the case where the maps have either rank 1 or rank 2.

A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp–Lieb inequality to hold. We present an algorithm which generates such a list.

Keywords

44A3514M1526D20Brascamp–Lieb inequalityLoomis–Whitney inequalitylatticeflag
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 4 , 01 August 2010 , pp. 870 - 888
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Barthe, F., On a reverse form of the Brascamp–Lieb inequality. Invent. Math. 134(1998), no. 2, 335–361. doi:10.1007/s002220050267Google Scholar
[2] Barvinok, A., A course in convexity. Graduate Studies in Mathematics, 54, American Mathematical Society, Providence, RI, 2002.Google Scholar
[3] Bennett, J., Carbery, A., Christ, M., and Tao, T., Finite bounds for Hölder–Brascamp–Lieb multilinear inequalities. To appear, Math. Res. Lett.Google Scholar
[4] Bennett, J., The Brascamp–Lieb inequalities: finiteness, structure and extremals. Geom. Funct. Anal. 17(2008), no. 5, 1343–1514. doi:10.1007/s00039-007-0619-6Google Scholar
[5] Brascamp, H. J. and Lieb, E. H., 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. doi:10.1016/0001-8708(76)90184-5Google Scholar
[6] Carlen, E. A., Lieb, E. H., and Loss, M., A sharp analog of Young's inequality on SN and related entropy inequalities. J. Geom. Anal. 14(2004), no. 3, 487–520.Google Scholar
[7] Lieb, E. H., Gaussian kernels have only Gaussian maximizers. Invent. Math. 102, no. 1, 179–208. doi:10.1007/BF01233426Google Scholar
[8] Rota, G.-C., The many lives of lattice theory. Notices Amer. Math. Soc. 44(1997), no. 11, 1440–1445.Google Scholar