On the property of four elements in modular spaces (Q1964624)

Let \((E, \Sigma, \mu)\) be a measurable space and let \(f:E\times \mathbb{R}_+\rightarrow \mathbb{R}_+\) be a function satisfying: (i) for \(\mu\) -- a.a. \(t\in E\), \(f(t,\cdot):\mathbb{R}_+ \rightarrow \mathbb{R}_+\) is a non-decreasing convex function such that \(f(t,0)= 0\) and \(f(t,u)>0\) if \(u>0\); (ii) \(f(\cdot, u): E\rightarrow \mathbb{R}_+\) is a \(\Sigma\)-measurable function for all \(u>0\). For a real (or complex) valued \(\Sigma\)-measurable function \(x=x(t)\) on \(E\), define the convex modular \(\rho_f\) by \(\rho_f(x) = \int_E f(t, |x(t)|) d\mu(t)\) and denote by \(X_{\rho_f}\) the corresponding Orlicz-Musielak space. If \(X_{\rho_f}\) is ordered by the cone \(K=\{x\in X_{\rho_f}: x \geq 0\}\), then every latticially closed subset \(D\) of \(X_{\rho_f}\) satisfies the property of four elements (with respect to the modular \(\rho_f\) and the cone \(K\)), that is, \[ \rho_f (x-w) + \rho (y-z) \geq \rho(x-z\vee w) + \rho(y-z\wedge w) \] for every \(x,y \in X_{\rho}\) such that \(x-y\in K\) and all \(w,z \in D\). This is one of the two main results of the paper. In the second theorem the authors consider a problem of the isotonicity of the metric projection operator in a modular space \(X_{\rho}\) and its connection with the property of four elements.