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.