On conditionally \(\delta\)-convex functions (Q532098)

Let \(X\) be a real vector space and let \(V\) be a subset of \(X\). Fix further a positive real number \(\delta\). A function \(f:V\to\mathbb R\) is said to be conditionally \(\delta\)-convex if, for each convex combination \(t_1v_1+\cdots+t_nv_n\) of elements of \(V\) such that \(t_1v_1+\cdots+t_nv_n\) belongs to \(V\), the following inequality holds true: \[ f(t_1v_1+\cdots+t_nv_n)\leq t_1f(v_1)+\cdots+t_nf(v_n)+\delta. \] The main result of the paper states that a function \(f:V\to\mathbb R\) is conditionally \(\delta\)-convex if and only if there exists a convex function \(\varphi:X\to[-\infty,+\infty[\) such that, for all elements \(v\in V\), \[ \varphi(v)\leq f(v)\leq\varphi(v)+\delta. \] As applications, some properties of conditionally \(\delta\)-convex functions are presented. In case \(X=\mathbb R^n\), the authors give further characterizations for a function to be conditionally \(\delta\)-convex.