In this paper we study the Poincare constant for the Gaussian measure re- stricted to D = R d B(y,r) where B(y,r) denotes the Euclidean ball with center y and radius r, and d � 2. We also study the case of the l ∞ 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.