Dual characterizations of three distance functions (Q6620710)

Let \((X,\left\Vert \cdot\right\Vert )\) be a real normed space and \((X^{\ast },\left\Vert \cdot\right\Vert _{\ast})\) be its dual space. The distance function to a closed convex set \(C\subset X\) is the function \(d_{C} :X\rightarrow\mathbb{R}\) defined by \(d_{C}(x)=\inf\left\{ \left\Vert x-c\right\Vert :c\in C\right\} ,\) while the oriented distance function associated to \(C\) is defined via the formula \(\Delta_{C}=d_{C}-d_{X\backslash C}.\) When \(C\) is convex and compact, one can also associate to \(C\) the farthest distance function \( F_{C}^{\left\Vert \cdot\right\Vert },\) defined by \(F_{C}^{\left\Vert \cdot\right\Vert }(x)=\max\left\{ \left\Vert x-c\right\Vert :c\in C\right\} .\) All these three functions are convex and the aim of the present paper is to characterize them in terms of Fenchel conjugates and Fenchel subdifferentials. \N\NGiven a lower semicontinuous proper convex function \(f:X\rightarrow\mathbb{R\cup\{\infty\}}\), we attach to it the function \(\eta_{f}:X^{\ast}\rightarrow\mathbb{R\cup\{\infty\}},\) defined by \(\eta_{f}(0)=0\) and \(\eta_{f}(x^{\ast})=\left\Vert x^{\ast}\right\Vert _{\ast }f^{\ast}\left( x^{\ast}/\left\Vert x^{\ast}\right\Vert \right) \) if \(x^{\ast}\neq0;\) here \(f^{\ast}\) denotes the Fenchel conjugate of \(f\). According to Corollary 2.4, for any function \(f:X\rightarrow\mathbb{R}\), the following statements are equivalent: (a)~there exists a nonempty closed convex set \(C\subset X\) such that \(f=d_{C};\) (b)~\(\partial f(x)\cap B^{\ast} \neq\emptyset\) for every \(x\in X\) and \(\eta_{f}(x^{\ast})=f^{\ast}(x^{\ast})\) for all \(x^{\ast}\in B^{\ast}.\) Here \(B^{\ast}=\{ x^{\ast}:x^{\ast} \in X^*\), \(\left\Vert x^{\ast}\right\Vert _{\ast}\leq1\} .\) In the context of Hilbert spaces, Theorem~3.1 asserts that the functions \(\Delta_{C}\) are precisely the functions \(f:X\rightarrow\mathbb{R}\) such that \(\partial f(x)\cap S^{\ast}\neq\emptyset\) for every \(x\in X\) and \(\eta_{f}\) is convex and lower semicontinuous. Here \(S^{\ast}=\{ x^{\ast}:x^{\ast} \in X^*\), \(\left\Vert x^{\ast}\right\Vert _{\ast}=1\} \). \N\NA characterization of the functions \(F_{C}^{\left\Vert \cdot\right\Vert }\) is provided by Theorem~4.1 in the case where \(X\) is the space \(\mathbb{R}^{N}\) endowed with a norm \(\left\Vert \cdot\right\Vert \) less than or equal to the Euclidean norm: For \(f:\mathbb{R}^{N}\rightarrow\mathbb{R}\), the following statements are equivalent: (a)~there exists a nonempty compact convex set \(C\subset\mathbb{R}^{N}\) such that \(f=F_{C}^{\left\Vert \cdot\right\Vert };\) (b)~\(\partial f(x)\cap S^{\ast}\neq\emptyset\) for every \(x\in\mathbb{R}^{N}\) and \(\eta_{f}\) is finite-valued and concave.