  In this paper we study the Poincar\'e constant for the Gaussian measure
restricted to $D=\R^d - B(y,r)$ where $B(y,r)$ denotes the Euclidean ball with
center $y$ and radius $r$, and $d\geq 2$. We also study the case of the
$l^\infty$ ball (the hypercube). This is the first step in the study of the
asymptotic behavior of a $d$-dimensional Ornstein-Uhlenbeck process in the
presence of obstacles with elastic normal reflections (the Ornstein-Uhlenbeck
pinball) we shall study in a companion paper.
