A Note on the Extension of Continuous Convex Functions from Subspaces

Let $Y$ be a subspace of a real normed space $X$. We say that the couple $(X,Y)$ has the $\mathrm{CE}$-property (``convex extension property'') if each continuous convex function on $Y$ admits a continuous convex extension defined on $X$. By using techniques of Johnson and Zippin, we prove the following results about the $\mathrm{CE}$-property: if $X$ is the $c_0(\Gamma)$-sum or the $\ell_p(\Gamma)$-sum ($1<p<\infty$) of separable normed spaces, then the couple $(X,Y)$ has the $\mathrm{CE}$-property, for each subspace $Y$ of $X$. Another similar result concerns weak$^*$-closed subspaces $Y$ of $X=\ell_1(\Gamma)=c_0(\Gamma)^*$.