Between quasi-convex and convex set-valued mappings. (Q1764501)

Let \(X\) be a linear space, \(D \subset X\) be a nonempty convex set, and let \(Y\) be a topological linear space partially ordered by a closed convex cone \(K\) with nonempty interior. A set-valued mapping \(F: D \rightarrow 2^Y\) is said to be \(K\)-convex if \(tF(x)+(1-t)F(x') \subset F(tx+(1-t)x')+K\) for all \(x,x' \in D\) and \(t \in [0,1]\). It is said to be \(K\)-quasi-convex if \(\{x \in D : y \in F(x) + K\}\) is convex for all \(y \in Y\). Any \(K\)-convex map is \(K\)-quasi-convex. In this paper, the authors give a sufficient condition for a quasi-convex map to be convex. The condition is that for every \(x \in D\), \(F(x)+K\) is closed and the \(K\)-convex condition holds on a nonempty compact set in \([0,1]\).