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Journal of Convex Analysis 30 (2023), No. 1, 329--342
Copyright Heldermann Verlag 2023

Rotundity Properties, and Non-Extendability of Lipschitz Quasiconvex Functions

Carlo Alberto De Bernardi
Dip. di Matematica per le Scienze Economiche, Finanziarie ed Attuariali, UniversitÓ Cattolica del Sacro Cuore, Milano, Italy

Libor Vesely
Dip. di Matematica "F. Enriques", UniversitÓ degli Studi di Milano, Italy

Let A be an open convex subset of a real normed linear space X. We prove that if the boundary of A contains a non-LUR point then there exists a Lipschitz quasiconvex function f from A to R not admitting any continuous quasiconvex extension to the whole X.

Keywords: Lipschitz function, quasiconvex function, LUR point, exposing functional.

MSC: 46A55, 52A05; 52A99, 26B25.

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