Open problems in \(\Phi\)-convexity (Q2772745)

The notion of \(\Phi\)-convexity was introduced in 1952 by \textit{J. W. Ellis} [cf. ``A general set-separation theorem'', Duke Math. J. 19, 417-421 (1952; Zbl 0047.28601)]. Let \(X\) be a set and \(\Phi\) be a class of real-valued functions defined on \(X\). The function \(f:X\to\overline{\mathbb R}\) is called \(\Phi\)-convex if for all \(x\in X\) NEWLINE\[NEWLINE f(x)=\sup\bigl\{ \varphi(x)+c: \varphi\in\Phi,\;c\in{\mathbb R}\text{ s.t. } \varphi(y)+c\leq f(y)\text{ for all \(y\in X\)} \bigr\}. NEWLINE\]NEWLINE Seventeen open problems concerning \(\Phi\)-convex analysis are presented. These problems are connected with: NEWLINENEWLINENEWLINE-- the classes of \(\Phi\)-convex functions for some special families \(\Phi\), NEWLINENEWLINENEWLINE-- subdifferentials of \(\Phi\)-convex functions as monotone multifunctions, NEWLINENEWLINENEWLINE-- uniqueness property, NEWLINENEWLINENEWLINE-- \(\Phi\)-separated sets, NEWLINENEWLINENEWLINE-- globalization (the problems when local notions are simultaneously global) for \(\Phi\)-convexity and \phantom{--} \(\Phi\)-subgradients, NEWLINENEWLINENEWLINE-- Lipschitz functions and Lipschitz multifunctions, NEWLINENEWLINENEWLINE-- relations between duality and Lipschitz functions, NEWLINENEWLINENEWLINE-- approximation of sets and decisively separated sets, NEWLINENEWLINENEWLINE-- Fréchet \(\Phi\)-differentiability of Lipschitz functions and monotonicity property.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00067].