We consider a variant of reachability in Vector Addition Systems (VAS) dubbed box reachability, whereby a vector v in N^d is box-reachable from 0 in a VAS V if V admits a path from 0 to v that not only stays in the positive orthant (as in the standard VAS semantics), but also stays below v, i.e., within the ׳׳box׳׳ whose opposite corners are 0 and v.
Our main result is that for two-dimensional VAS, the set of box-reachable vertices almost coincides with the standard reachability set: the two sets coincide for all vectors whose coordinates are both above some threshold W. We also study properties of box-reachability, exploring the differences and similarities with standard reachability.
Technically, our main result is proved using powerful machinery from convex geometry.