2022/12/28 by Carlo Alberto De Bernardi, De Bernardi, Carlo Alberto, Libor Veselý +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2212.13789
openalex publication_date 2022/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 \citeDEQEX to the infinite-dimensional separable case. Moreover, we also immediately obtain existence of examples in which (QE) does not hold.