The weak separation in higher dimensions

For an odd integer r>0 and n>r, we introduce a notion of weakly r-separated collections subsets [n]={1,2,…,n}. When r=1, this corresponds to the concept weak separation introduced by Leclerc Zelevinsky. In paper, extending results due Leclerc-Zelevinsky, develop geometric approach establish number nice combinatorial properties maximal (such as exact upper bound on size collections, mutations rules, relations so-called membranes in zonotopes dimension r+2, etc.) A possible analog with r even is briefly discussed well.