Extendability of continuous quasiconvex functions from subspaces

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