Explicit convex and concave envelopes through polyhedral subdivisions

5 In this paper, we derive explicit characterizations of convex and concave envelopes of several 6 nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained 7 by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be 8 constructed easily. In particular, we use these techniques to derive, in closed-form, the concave 9 envelopes of concave-extendable supermodular functions and the convex envelopes of disjunctive 10 convex functions. 11