On the Extension of Continuous Quasiconvex Functions
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On the Extension of Continuous Quasiconvex Functions Carlo Alberto De Bernardi1 Received: 10 July 2020 / Accepted: 6 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study the problem of extending continuous quasiconvex real-valued functions from a subspace of a real normed linear space. Our results are essentially finite-dimensional and are based on a technical lemma which permits to “extend” a nested family of open convex subsets of a given subspace to a nested family of open convex sets in the whole space, in such a way that certain topological conditions are satisfied. Keywords Quasiconvex function · Extension · Convex set · Normed linear space Mathematics Subject Classification Primary 52A41; Secondary 26B25 · 46A99
1 Introduction Let X be a nontrivial real normed linear space, Y a subspace of X , and A a convex subset of X . A function f : A → R is called quasiconvex if its lower level sets are convex. Characteristic properties of functions with convex lower level sets were considered for the first time by De Finetti [1], although Fenchel was the first in naming and formalizing quasiconvex functions in [2]. Quasiconvex functions are a natural generalization of convex functions and play a crucial role in mathematical programming, in economics, and in many other areas of mathematical analysis (see [3–6] and the references therein). Moreover, the well-known result by Crouzeix contained in [7] (see also [8]) and the recent results contained in [9–12] show that quasiconvex functions satisfy remarkable continuity and differentiability properties. The natural problem of extending continuous convex real-valued functions from Y to the whole X was studied by several authors in the last two decades (see, e.g. [13–18] and the references therein), and positive results were obtained under suitable
Communicated by Nicolas Hadjisavvas.
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Carlo Alberto De Bernardi [email protected]; [email protected] Dipartimento di Matematica per le Scienze economiche, finanziarie ed attuariali, Università Cattolica del Sacro Cuore, Via Necchi 9, 20123 Milan, Italy
123
Journal of Optimization Theory and Applications
hypotheses involving separability conditions. In [16,17], the authors consider the more general problem concerning extendability of continuous convex functions from A ∩ Y to A, where A ⊂ X is an open convex set intersecting Y . The aim of the present paper is to start the study of the corresponding problem for the class of continuous quasiconvex functions. Our results are essentially finite-dimensional, the investigation of the infinite-dimensional case will be the subject of a subsequent study. The paper is organized as follows. After some preliminaries, contained in Sect. 2, in Sect. 3 we present our main result, namely Theorem 3.1, stating that if X is finitedimensional then every continuous quasiconvex function defined on A ∩ Y admits a continuous quasiconvex extension defined on the whole A. The proof of this result is based on the follo
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