Non polyhedral convex envelopes for 1-convex functions

In this paper we discuss how to derive the non polyhedral convex envelopes for some functions, called 1-convex throughout the paper, over boxes. The main result is about n-dimensional 1-convex functions, but we get to it by first discussing in detail some special cases, namely functions (Formula presented.), (Formula presented.), and, next, more general trivariate functions. The relation between the class of functions investigated in this paper and other classes investigated in the existing literature is discussed.