In R^2, for v(x,y), the condition can be written so the corresponding region is a half plane. (bounded below* by the straight line ) Similarly, if u_1 or u_2=0 than the region is either a half plane, either the whole plane, either the empty set : even if the regions are various, it's always the same kind of domain : a (half) plane.
To answer the first question, you can proceed this way : For any two points v and w in S and for any real number t in ]0,1[, show that t*v+(1-t)*w is in S that is to say that .
Does it help ?
(*) : Is their a more suitable expression than "bounded below" that could be used in this case ?