Duality gap of the conic convex constrained optimization problems in normed spaces

In this paper, motivated by a result due to Champion [Math. Program. 99, 2004], we introduce a property D(y) for a conic quasi-convex vector-valued function in a general normed space. We prove that this property D(y) characterizes the zero duality gap for a class of the conic convex constrained optimization problem in the sense that if this property is satisfied and the objective function f is continuous at some feasible point, then the duality gap is zero, and if this property is not satisfied, then there exists a linear continuous function f such that the duality gap is positive. We also present some sufficient conditions for the property D(y).