I am interested in isoperimetric equalities in lattices, more precisely what could be said for surrounding territory in the game of go. Here are some statements or heuristics:
To surround territory with a certain number of stones, it is better to do squares than thin rectangles: this is a basic isoperimetric inequality/optimization problem, not particularly combinatoric.
Using edges is better than being in the center: the edges essentially "turn into friend stones" of neighboring territories, saving lots of moves to surround territory. Using corners is even better.
However there are obviously finer considerations: what is the optimal shape to make territory (the circle would be in the Euclidian plane, but here we are on a grid)? What about the use of edges/corners: the better shape there may be different (a straight line looks more efficient than half a square for instance).
Are there similar questions on more general lattices maybe? Any reference is welcome.