A Property of Strictly Convex Functions which Differ from each other by a Constant on the Boundary of their Domain

We prove, in particular, the following result: Let $E$ be a reflexive real Banach space and let $C\subset E$ be a closed convex set, with non-empty interior, whose boundary is sequentially weakly closed and non-convex. Then, for every function $\varphi:\partial C\to {\bf R}$ and for every convex set $S\subseteq E^*$ dense in $E^*$, there exists $\tilde\gamma\in S$ having the following property: for every strictly convex lower semicontinuous function $J:C\to {\bf R}$, G\^ateaux differentiable in $\hbox {\rm int}(C)$, such that $J_{|\partial C}-\varphi$ is constant in $\partial C$ and $\lim_{\|x\|\to +\infty}\,(J(x)/\|x\|) = +\infty$ if $C$ is unbounded, $\tilde\gamma$ is an algebraically interior point of $J'(\hbox {\rm int}(C))$ (with respect to $E^*$).