We consider a Gibbs sampler applied to the uniform distribution on a bounde
d region R subset of or equal to R-d. We show that the convergence properti
es of the Gibbs sampler depend greatly on the smoothness of the boundary of
R. Indeed, for sufficiently smooth boundaries the sampler is uniformly erg
odic, while for jagged boundaries the sampler could fail to even be geometr
ically ergodic.