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

be a reflexive real Banach space and let

C\subset E C ⊂ 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} φ : ∂ C → R

and for every convex set

S\subseteq E^* S ⊆ E ∗

dense in

E^* E ∗

, there exists

\tilde\gamma\in S γ ~ ∈ S

having the following property: for every strictly convex lower semicontinuous function

J:C\to {\bf R} J : C → R

, Gâteaux differentiable in

\hbox {\rm int}(C) int ( C )

, such that

J_{|\partial C}-\varphi J ∣ ∂ C − φ

is constant in

\partial C ∂ C

and

\lim_{\|x\|\to +\infty}\,(J(x)/\|x\|) = +\infty lim ⁡ ∥ x ∥ → + ∞   ( J ( x ) / ∥ x ∥ ) = + ∞

if

C C

is unbounded,

\tilde\gamma γ ~

is an algebraically interior point of

J'(\hbox {\rm int}(C)) J ′ ( int ( C ) )

(with respect to

E^* E ∗

).