Generalization of Asplund inequalities on Lipschitz functions (Q1312343)

Let \((X,d)\) be a metric space. Let \(\Phi\) be a certain family of Lipschitz functions defined on \(X\). We assume that \(\Phi+ c=\Phi\) for \(c\in R\). We denote \[ f^*(\phi)=\sup_{x\in X}\bigl[\phi(x)- f(x)\bigr].\tag{1} \] The function \(f^*(\phi)\) defined on \(\Phi\) will be called Fenchel dual function, (or Fenchel conjugate function). We say that a function \(f\) is \(\Phi\)-convex if it is a majoring of the functions of \(\Phi\), \(f(x)=\sup\bigl\{\phi(x): \phi\in \Phi, \phi\leq f\bigr\}\). By \(d_ L(\phi_ 1,\phi_ 2)\) we shall denote the infimum of Lipschitz constants of the function \(\phi_ 1-\phi_ 2\). Having this notation we can extend the well-known Asplund Theorem [see \textit{E. Asplund}, Acta Math. 121, 31-47 (1968; Zbl 0162.175)] to Lipschitz functions. Theorem 1. Let \(f(x)\) be a \(\Phi\)-convex function. Suppose that \[ f^*(\phi)\geq f^*(\phi_ 0)_ \phi(x_ 0)- \phi_ 0(x_ 0)+ \gamma(d_ L(\phi,\phi_ 0))\tag{2} \] holds. Then \[ f(x)\leq f(x_ 0)+ \phi_ 0(x)- \phi_ 0(x_ 0)+ \gamma^*(d(x,x_ 0)),\tag{3} \] where \(\gamma^*(t)\) is a dual function to the function \(\gamma\), \(\gamma^*(t)=\sup_{u>0} [ut-\gamma(u)]\). The proof of the converse implication request additional conditions on the space \(X\) and on the class \(\Phi\).