  This paper presents a detailed theoretical analysis of the Langevin Monte
Carlo sampling algorithm recently introduced in Durmus et al. (Efficient
Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets
Moreau, 2016) when applied to log-concave probability distributions that are
restricted to a convex body $\mathsf{K}$. This method relies on a
regularisation procedure involving the Moreau-Yosida envelope of the indicator
function associated with $\mathsf{K}$. Explicit convergence bounds in total
variation norm and in Wasserstein distance of order $1$ are established. In
particular, we show that the complexity of this algorithm given a first order
oracle is polynomial in the dimension of the state space. Finally, some
numerical experiments are presented to compare our method with competing MCMC
approaches from the literature.
