We propose a new concept of S-convex function (and its variant SSQS-convexity) which includes M-natural-convex function as a subclass, and establish its fundamental properties in continuous spaces. A characterization of a twice continuously differentiable S-convex function is provided using its Hessian. Several preservation properties of S-convexity are presented. We show that S-convex functions form a subclass of supermodular functions which have a one-to-one correspondence with jointly submodular and convex functions through the conjugate operator under mild conditions. Under the same conditions, we build the equivalence between S-concavity and an important concept of gross substitutability in economics. In addition, we prove that coercive M-natural-concavity is equivalent to a stronger version of gross substitutability. In a parametric maximization model with a box constraint, we show that the set of optimal solutions is nonincreasing in the parameters if the objective function is (SSQS-)S-concave.Our theoretical results are applied to two notable operations models. In a random yield inventory model where a buyer procures a single product from multiple unreliable suppliers to fulfill its random demand, we provide conditions under which the ordering quantity vector is nonincreasing in the initial inventory. For a classical multi-product inventory model, we show that S-convexity leads to two desired decreasing properties in the literature, which includes several results in the literature as special cases and significantly simplifies the analysis
