Generate samples by the Metropolis-Hastings algorithm which have a minimum distance between one another Can we generate samples by the Metropolis-Hastings algorithm with target density $p(x)$ and proposal kernel density $q(x,y)$ such that the samples satisfy a certain constraint?
Take the simple example where the sample space is $[0,1)^2$. Can we generate samples such that the sample points have a given minimal distance?
I think the only way to do this is ensuring that the proposals satisfy this constraint, but please let me know if I'm wrong. Since this means we need ensure that samples distributed according to $q(x,\;\cdot\;)$ satisfy this constraint, I think the question boils down to asking whether we can sample from any distribution and ensure that the generated samples satisfy the constraint (e.g. have a certain minimal distance).
If that's correct at all, how should we do that? I think the most simple approach would be to generate each sample in a loop and only accepting it, if the constraint is satisfied. But maybe there is a better option.

This constraint is contradictory with the aim of generating marginal realisations from the target density. Under stationarity, Metropolis-Hastings produces (Markov) dependent simulations such that each term in the sequence is distributed from $p(\cdot)$. It does not aim at a specified joint distribution of the chain (even though its joint distribution obviously exists by the very construction of the algorithm).
To achieve a minimal distance between realisations is not achievable this way since the minimal distance constraint is expressed in terms of the joint distribution of the chain. This seems to require low discrepancy sequences used in quasi-Monte Carlo Metropolis algorithms such as Owen's (2005).
