  For domains in $\mathbb{R}^d$, $d\geq 2$, we prove universal upper and lower
bounds on the product of the bottom of the spectrum for the Laplacian to the
power $p>0$ and the supremum over all starting points of the $p$-moments of the
exit time of Brownian motion. It is shown that the lower bound is sharp for
integer values of $p$ and that for $p \geq 1$, the upper bound is
asymptotically sharp as $d\to\infty$. For all $p>0$, we prove the existence of
an extremal domain among the class of domains that are convex and symmetric
with respect to all coordinate axes. For this class of domains we conjecture
that the cube is extremal.
