Ornstein-Uhlenbeck pinball: I. Poincar\'e inequalities in a punctured domain

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.