Abstract

Let XX be a topological vector space, Y⊂XY\subset X a subspace, and A⊂XA\subset X an open convex set containing 00. We are interested in the extendability of a continuous convex function f ⁣:A∩Y→Rf\colon A\cap Y\to\mathbb{R} to a continuous convex function F ⁣:A→RF\colon A\to\mathbb{R}. We characterize such extendability: (a) for a given ff; (b) for every ff. The case (b) for A=XA=X generalizes results from a paper by J. Borwein, V. Montesinos and J. Vanderwerff [Boundedness, differentiability and extensions of convex functions, J. Convex Analysis 13 (2006) 587–602], and from another one by L. Zajíček and the second author [On extensions of d.c. functions and convex functions, J. Convex Analysis 17 (2010) 427–440]. We also show that if XX is locally convex and X/YX/Y is ``conditionally separable'', then the couple (X,Y)(X,Y) satisfies the CE\mathrm{CE}-property, saying that the above extendability holds for A=XA=X and every ff. It follows that every couple (X,Y)(X,Y) has the CE\mathrm{CE}-property for the weak topology. We consider also a stronger SCE\mathrm{SCE}-property saying that the above extendability is true for every AA and every ff. A deeper study of the SCE\mathrm{SCE}-property will appear in a subsequent paper.

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, 46A99

References

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Suggested citation

C. A. De Bernardi, L. Veselý. “Extension of Continuous Convex Functions from Subspaces I.” Journal of Convex Analysis 21 (2014), No. 4, 1065–1084. https://doi.org/10.68381/jca21057

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.