A characterization of essentially strictly convex functions on reflexive Banach spaces
Let \(X\) be a reflexive Banach space with dual \(X^*\), let \(J: X\to\mathbb{R}\cup\{\infty\}\) be a properly (not necessarily convex) lower-semicontinuous function with conjugate \(f^*\) and subdifferential mapping \(\partial f\), and let \[ \begin{aligned} MJ(x^*) &:= (\partial f)^{-1}(x^*)\\ &=\text{argmin}\{f(y)-\langle x^*,y\rangle\mid y\in\text{dom\,}f\}\\ &=\{x\in X\mid\langle x^*,y\rangle= f(x)+ f^*(x^*)\}\\ &\subset\partial f^*(x^*).\end{aligned} \] On the basis of the differentiability properties of conjugate functions it is shown that \(MJ\) is single-valued on \(\text{dom}(\partial f^*)\) if and only if \(f\) is essentially strictly convex. The result is specified for functions of the form \(J= f+ h\) where \(f\) is a convex function with \(\text{dom\,}f^*= X^*\) and \(h\) a (not necessarily convex) function with \(\text{dom\,}h^*\neq\emptyset\). If \(f(x)= 1/2\| x\|^2\) then \(MJ= \text{Proc}^f_h\) is the so-called Moreau proximal set-valued mapping associated with \(h\) and \(J^*= (f+ h)^*=\Phi^f_h\) can be viewed as an extension of the Asplund function. Conversely, setting \(h= i_S\) (the indicator function of a set \(S\subset X\)) then an assertion concerning Bregman-Tschebychev sets and functions with respect to the function \(f\) can be derived.
scientific article; zbMATH DE number 5991163
scientific article