Abstract

Given YY a subspace of a topological vector space XX, and an open convex set 0∈A⊂X0\in A\subset X, we say that the couple (X,Y)(X,Y) has the CE(A)\mathrm{CE}(A)-property if each continuous convex function on A∩YA\cap Y admits a continuous convex extension defined on AA. Using results from our previous paper, we study for given AA the relation between the CE(A)\mathrm{CE}(A)-property and the CE(X)\mathrm{CE}(X)-property. As a corollary we obtain that (X,Y)(X,Y) has the CE(A)\mathrm{CE}(A)-property for each AA, provided (X,Y)(X,Y) has the CE(X)\mathrm{CE}(X)-property and YY is ``conditionally separable''. This applies, for instance, if XX is locally convex and conditionally separable. Other results concern either the CE(A)\mathrm{CE}(A)-property for sets AA of special forms, or the CE(A)\mathrm{CE}(A)-property for each AA where XX is a normed space with X/YX/Y separable. In the last section, we point out connections between the CE(X)\mathrm{CE}(X)-property and extendability of certain continuous linear operators. This easily yields a generalization of an extension theorem of Rosenthal, and another result of the same type.

Author information

Contact details are reproduced from the original publication and may be historical.

Libor Veselý

Dipartimento di Matematica, Università degli Studi, Via C. Saldini 50, 20133 Milano, Italy

libor.vesely@unimi.it

Keywords

  • Convex function
  • extension
  • topological vector space
  • normed linear space

Mathematics Subject Classification

52A41; 26B25, 47A99

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Suggested citation

C. A. De Bernardi, L. Veselý. “Extension of Continuous Convex Functions from Subspaces II.” Journal of Convex Analysis 22 (2015), No. 1, 101–116. https://doi.org/10.68381/jca22006

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.