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# The Brascamp-Lieb Polyhedron

Published:2010-05-20
Printed: Aug 2010
• Stefán Ingi Valdimarsson,
UCLA Mathematics Department, Los Angeles, CA, USA
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## Abstract

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: Brascamp-Lieb inequality, Loomis-Whitney inequality, lattice, flag
 MSC Classifications: 44A35 - Convolution 14M15 - Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 26D20 - Other analytical inequalities

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