On convergence rates of Gibbs samplers for uniform distributions

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.