The group problem on the unit interval is developed, with and without continuous variables. The connection with cutting planes, or valid inequalities, is reviewed. Certain desirable properties of valid inequalities, such as minimality and extremality are developed, and the connection between valid inequalities for P(I, u0) and P-+(I, u0) is developed. A class of functions is shown to give extreme valid inequalities for P-+(I, u0) and for certain subsets U of I. A method is used to generate such functions. These functions give faces of certain corner polyhedra. Other functions which do not immediately give extreme valid inequalities are altered to construct a class of faces for certain corner polyhedra. This class of faces grows exponentially as the size of the group grows. © 1972 The Mathematical Programming Society.