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In this paper, in particular, we prove the following result: Let $E$ be areflexive real Banach space and let $C\subset E$ be a closed convex set, withnon-empty interior, whose boundary is sequentially weakly closed andnon-convex. Then, for every function $\varphi:\partial C\to {\bf R}$ and forevery convex set $S\subseteq E^*$ dense in $E^*$, there exists $\tilde\gamma\inS$ having the following property: for every strictly convex lowersemicontinuous function $J:C\to {\bf R}$, G\^ateaux differentiable in $\hbox{int}(C)$, such that $J_{|\partial C}-\varphi$ is constant in $\partial C$ and$\lim_{\|x\|\to +\infty}{{J(x)}\over {\|x\|}}=+\infty$ if $C$ is unbounded,$\tilde\gamma$ is an algebraically interior point of $J'(\hbox {int}(C))$ (withrespect to $E^*$).

A property of strictly convex functions which differ from each other by a constant on the boundary of their domain