On convergence rates of Gibbs samplers for uniform distributions

We consider a Gibbs sampler applied to the uniform distribution on a bounded region R.Rd. We show that the convergence properties of the Gibbs sampler depend greatly on the smoothness of the boundary of R. Indeed, for sufficiently smooth boundaries the sampler is uniformly ergodic, while for jagged boundaries the sampler could fail to even be geometrically ergodic.