A Note on the Extension of Continuous Convex Functions from Subspaces

Let

be a subspace of a real normed space

X X

. We say that the couple

(X,Y) ( X , Y )

has the

\mathrm{CE} C E

-property (“convex extension property”) if each continuous convex function on

Y Y

admits a continuous convex extension defined on

X X

. By using techniques of Johnson and Zippin, we prove the following results about the

\mathrm{CE} C E

-property: if

X X

is the

c_0(\Gamma) c 0 ( Γ )

-sum or the

\ell_p(\Gamma) ℓ p ( Γ )

-sum (

1<p<\infty 1 < p < ∞

) of separable normed spaces, then the couple

(X,Y) ( X , Y )

has the

\mathrm{CE} C E

-property, for each subspace

Y Y

of

X X

. Another similar result concerns weak

^* ∗

-closed subspaces

Y Y

of

X=\ell_1(\Gamma)=c_0(\Gamma)^* X = ℓ 1 ( Γ ) = c 0 ( Γ ) ∗

.