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

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.