  In this paper, we explore sampling from strongly log-concave distributions
defined on convex and compact supports. We propose a general proximal framework
that involves projecting onto the constrained set, which is highly flexible and
supports various projection options. Specifically, we consider the cases of
Euclidean and Gauge projections, with the latter having the advantage of being
performed efficiently using a membership oracle. This framework can be
seamlessly integrated with multiple sampling methods. Our analysis focuses on
Langevin-type sampling algorithms within the context of constrained sampling.
We provide nonasymptotic upper bounds on the W1 and W2 errors, offering a
detailed comparison of the performance of these methods in constrained
sampling.
