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15 May 2002 Brascamp-Lieb-Luttinger inequalities for convex domains of finite inradius
Pedro J. Méndez-Hernández
Duke Math. J. 113(1): 93-131 (15 May 2002). DOI: 10.1215/S0012-7094-02-11313-1


We prove a multiple integral inequality for convex domains in $\mathbf {R}\sp n$ of finite inradius. This inequality is a version of the classical inequality of H. Brascamp, E. Lieb, and J. Luttinger, but here, instead of fixing the volume of the domain, one fixes its inradius $r\sb D$ and the ball is replaced by $(-r\sb D, r\sb D)\times $\mathbf {R}\sp {n-1}$. We also obtain a sharper version of our multiple integral inequality, which generalizes the results in [6], for two-dimensional bounded convex domains where we replace infinite strips by rectangles. It is well known by now that the Brascamp-Lieb-Luttinger inequality provides a powerful and elegant method for obtaining and extending many of the classical geometric and physical isoperimetric inequalities of G. Pólya and G. Szegö. In a similar fashion, the new multiple integral inequalities in this paper yield various new isoperimetric-type inequalities for Brownian motion and symmetric stable processes in convex domains of fixed inradius which refine in various ways the results in [2], [3], [4], [5], and [20]. These include extensions to heat kernels, heat content, and torsional rigidity. Finally, our results also apply to the processes studied in [10] whose generators are relativistic Schrödinger operators.


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Pedro J. Méndez-Hernández. "Brascamp-Lieb-Luttinger inequalities for convex domains of finite inradius." Duke Math. J. 113 (1) 93 - 131, 15 May 2002.


Published: 15 May 2002
First available in Project Euclid: 18 June 2004

zbMATH: 1009.31003
MathSciNet: MR1905393
Digital Object Identifier: 10.1215/S0012-7094-02-11313-1

Primary: 31B35

Rights: Copyright © 2002 Duke University Press


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Vol.113 • No. 1 • 15 May 2002
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