Extendability of continuous quasiconvex functions from subspaces

Let Y be a subspace of topological vector space X, and A⊂X an open convex set that intersects Y. We say the property (QE) [property (CE)] holds if every continuous quasiconvex [continuous convex] function on A∩Y admits extension defined A. study relations between (CE) properties, proving always implies that, under suitable hypotheses (satisfied for example X is normed closed X), two properties are equivalent. By combining previous implications with known results about (CE), we obtain some new positive functions. In particular, generalize contained in [9] to infinite-dimensional separable case. Moreover, also immediately existence examples which does not hold.