The relative interior of the base polyhedron and the core

This note provides two necessary and sufficient conditions for the relative interior of the core (and the base polyhedron) to be non-empty: (i) the se cond largest excess of the prenucleolus is negative; (ii) the grand coaliti on's payoff is greater than the minimum no-blocking payoff. Such conditions imply an intuitive method in proving core existence, they also imply resul ts on the sensitivity of the base polyhedron and the core.