Optimal structural policies for ambiguity and risk averse inventory and pricing models: Online Appendix

Lemma A.1 (a) A real-valued convex function is also 0-convex and hence k-convex for all k ≥ 0. A k1-convex function is also a k2-convex function for k1 ≤ k2. (b) If f1(y) and f2(y) are k1-convex and k2-convex respectively, then for α, β ≥ 0, αf1(y) + βf2(y) is (αk1 + βk2)-convex. (c) If f(y) is k-convex and w is a random variable, then E{f(y − w)} is also k-convex, provided E{|f(y − w)|} < ∞ for all y. (d) Assume that f is a continuous k-convex function and f(y) → ∞ as |y| → ∞. Let S be a minimum point of g and s be any element of the set