The nucleolus is well posed

The lexicographic order is not representable by a real-valued function, contrary to many other
orders or preorders. So, standard tools and results for well-posed minimum problems cannot be used.
We prove that under suitable hypotheses it is however possible to guarantee the well-posedness of a
lexicographic minimum over a compact or convex set. This result allows us to prove that some game
theoretical solution concepts, based on lexicographic order are well-posed: in particular, this is true
for the nucleolus.