So I'm currently trying to implement an MCMC to evenly sample hyperpoints from the polytope that is defined as $ mathbb {K} = {x in mathbb {R} ^ {n} ; ; text {s.t.} ; ; A , x = b } $ in the special case where there is a generic linear transformation $ A in mathbb {R} ^ {m times n} $. $ b equiv0 in mathbb {R} ^ m $ and the boundary conditions $ 0 leq x leq1 $ stop.
Although I was able to successfully run the simulation (I'm using Julia), there are a few things I'm not sure about:

Given that polytopes of this type tend to have starshaped shapes in larger dimensions, two preprocessing steps are required before starting the simulation:

The first concerns the socalled Blocked river setting That is to find quotation from the text of the exercise, Flux $ i $ so that $ max_ {x in K} x_i = min_ {x in K} x_i = z_i $ and remove such variables from the system by adjusting the vector $ b $, Can someone please explain what the hell that means?

The second is to find an optimal inner point $ mathbb {K} $ as the starting point of the chain, which must be intuitively far from the vertices of the polytope. The text tells me the following: e.g. by calculating $ frac {1} {2n} sum_ {i = 1} ^ n (x ^ { min, i} + x ^ { max, i}) $ Where $ x ^ { min, i} in arg min_ {x in K} x_i $ and $ x ^ { max, i} in arg max_ {x in K} x_i $, Here I simply don't understand the notation: I suppose I should calculate a weighted average of the centers of each edge of the polytope, but I cannot see how this is related to the above wording.


as with any coherent mcmc, the walk in the state space must be at t. it satisfies the detailed balance sheetIn this case, the text says that the goal should be distributed $$ p (x) propto delta ^ m (Sxb) prod_i theta (u_ix_i) theta (x_il_i) $$
Again, I have no idea how to get this and not how to calculate me.