## Duke Mathematical Journal

### Brascamp-Lieb-Luttinger inequalities for convex domains of finite inradius

Pedro J. Méndez-Hernández

#### Abstract

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.

#### Article information

Source
Duke Math. J. Volume 113, Number 1 (2002), 93-131.

Dates
First available in Project Euclid: 18 June 2004

http://projecteuclid.org/euclid.dmj/1087575226

Digital Object Identifier
doi:10.1215/S0012-7094-02-11313-1

Mathematical Reviews number (MathSciNet)
MR1905393

Zentralblatt MATH identifier
1009.31003

Subjects
Primary: 31B35: Connections with differential equations

#### Citation

Méndez-Hernández, Pedro J. Brascamp-Lieb-Luttinger inequalities for convex domains of finite inradius. Duke Math. J. 113 (2002), no. 1, 93--131. doi:10.1215/S0012-7094-02-11313-1. http://projecteuclid.org/euclid.dmj/1087575226.

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