Generalized quasiconvexity via properly characteristic functions associated to binary relations (Q5946856)

Let \(E_{1}\) and \(E_{2}\) be two nonempty sets, and let \(\Gamma :E_{1}\times E_{1}\to 2^{E_{1}}\) and \(\Omega :E_{2}\to 2^{E_{2}}.\) A subset \(X\) of \(E_{1}\) is said to be \(\Gamma\)-convex if \(\Gamma ( x^{1},x^{2}) \subset X,\) \(\forall x^{1},x^{2}\in X.\) A function \(f:X\to E_{2}\) defined on a nonempty and \(\Gamma\)-convex subset \(X\) of \(E_{1}\) is called \(( \Gamma ,\Omega)\)-quasiconvex if \(\forall x^{1},x^{2}\in X\), \(\forall y\in E_{2}\), \(f(\{ x^{1},x^{2}\}) \subset \Omega y \Rightarrow f( \Gamma ( x^{1},x^{2})) \subset \Omega y.{1}\) Given a real number \(\lambda\), a function \(g:E_{2}\times E_{2}\to{\mathbb R}\) is called \(\lambda\)-characteristic for \(\Omega \) if, for any \(y^{1},y^{2}\in E_{2}\to {\mathbb R},\) one has \(g( y^{1},y^{2}) \leq \lambda \Leftrightarrow y^{1}\in \Omega y^{2}.{1}\) This paper characterizes \(( \Gamma ,\Omega)\)-quasiconvexity in terms of scalar quasiconvexity, by using certain characteristic functions.