When some variational properties force convexity (Q2842249)

For a Banach space \(X\) denote by \(F(X)\) the set of all functions \(J:X\to\mathbb R\cup\{+\infty\}\) with dom\,\(J:=\{x\in X : J(x)<\infty\}\neq \emptyset.\) For \(J\in F(X)\) let \(\partial J,\, J^*, J^{**}\) be its subdifferential, its Legendre-Fenchel conjugate, and its biconjugate (restricted to \(X\)), respectively. Put also \(MJ=(\partial J)^{-1}:X^*\rightrightarrows X\) and \(\Gamma(X)=\{J\in F(X) : J=J^{**}\}.\) A function \(H\in \Gamma(X)\) is called essentially strictly convex (essentially Gâteaux differentiable) if \(MH\) is locally bounded and \(H\) is strictly convex on line segments in dom\,\(\partial H\) (respectively, dom\,\(\partial H\) is open and \(\partial H\) is single-valued on dom\, \(\partial H\)), see [\textit{H. H. Bauschke, J. M. Borwein} and \textit{P. L. Combettes}, Commun. Contemp. Math. 3, No. 4, 615--647 (2001; Zbl 1032.49025)]. \textit{M. Volle} and \textit{J.-B. Hiriart-Urruty} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 3, 1617--1622 (2012; Zbl 1244.46019)] have shown that, if \(X\) is reflexive, then \(H\in \Gamma(X)\) is essentially strictly convex if and only if \(J^*\) is essentially Gâteaux differentiable. The authors consider several conditions, labelled \((A), (A_0) , A^+_s, A^+_w \) (\(s\) for ``strong'' and \(w\) for ``weak''), ensuring that the lower semicontinuous hull \(\bar J\) of a function \(J\in F(X)\) is essentially strictly convex. The main result of the paper (Theorem 3.1) asserts that, if \(J\in F(X)\) satisfies \((A_0)\) (i.e., dom\,\(MJ =\) dom\,\(\partial J^*\) is nonempty and open) and \(J^*\) is essentially Gâteaux differentiable, then \(\bar J=J^{**}\) and \(\bar J\) is essentially strictly convex.NEWLINENEWLINESome interesting applications to the convexity of Chebyshev subsets of a Hilbert space are given. For instance, if \(S\) is a nonempty subset of a Hilbert space \(X\), then the following are equivalent: (i) for every \(u\in X\), the problem \(\min_{x\in S}\|x-u\|\) is Tychonov well posed; (ii) for every \(u\in X\), the problem \(\min_{x\in S}\|x-u\|\) is weakly Tychonov well posed; (iii) the set \(S\) is closed and convex; (iv) \(S\) is a Chebyshev set and the metric projection \(p_S\) is norm-to-norm continuous; (iv) \(S\) is a Chebyshev set and the metric projection \(p_S\) is norm-to-weak continuous (Corollary 3.8).NEWLINENEWLINE The equivalent problem, namely, that the only uniquely remotal sets in Hilbert space are the singletons, is discussed as well. As is known, the problem of the convexity of Chebyshev subsets of a Hilbert space is still open, see [\textit{J. M. Borwein}, Optim. Lett. 1, No. 1, 21--32 (2007; Zbl 1138.46009)] for a recent discussion.