Extension of Continuous Convex Functions from Subspaces I

Let

be a topological vector space,

Y\subset X Y ⊂ X

a subspace, and

A\subset X A ⊂ X

an open convex set containing

0 0

. We are interested in the extendability of a continuous convex function

f\colon A\cap Y\to\mathbb{R} f  ⁣ : A ∩ Y → R

to a continuous convex function

F\colon A\to\mathbb{R} F  ⁣ : A → R

. We characterize such extendability: (a) for a given

f f

; (b) for every

f f

. The case (b) for

A=X A = 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

X X

is locally convex and

X/Y X / Y

is “conditionally separable”, then the couple

(X,Y) ( X , Y )

satisfies the

\mathrm{CE} C E

-property, saying that the above extendability holds for

A=X A = X

and every

f f

. It follows that every couple

(X,Y) ( X , Y )

has the

\mathrm{CE} C E

-property for the weak topology. We consider also a stronger

\mathrm{SCE} S C E

-property saying that the above extendability is true for every

A A

and every

f f

. A deeper study of the

\mathrm{SCE} S C E

-property will appear in a subsequent paper.