Another isometry for the Legendre-Fenchel transform
Given a reflexive Banach space X and three numbers \(p\in]1,+\infty [\), \(\lambda >0\), \(\rho\geq 0\), the authors introduce the function (defined for \(x\in X\) and \(v\in X')\) \[ f_{\lambda}^{\#,p}(x,v)=\inf_{u\in X}[f(x+u)+\frac{1}{p\lambda}\| u\|^ p-<v,u>] \] and the distance \[ d_{\lambda,\rho}^{\#,p}(f,g)=\sup_{\| x\| \leq \rho, \| v\| \leq \rho}| f_{\lambda}^{\#,p}(x,v)- g_{\lambda}^{\#,p}(x,v)|, \] where f and g are two proper convex lower semicontinuous functions on X. If we denote by \(f\to f^*\) the Legendre-Fenchel transform, the main results of the paper are: a) The function \((x,v)\to f_{\lambda}^{\#,p}(x,v)\) is convex-concave, continuous, finite valued, and bounded on bounded subsets of \(X\times X'.\) b) If \(1/p+1/p'=1\) and \(\lambda '=\lambda^{-p'/p}\) we have \[ (f^*)_{\lambda}^{\#,p}(v,x)+f_{\lambda '}^{\#,p'}(x,v)=<v,x>. \] c) \(d_{\lambda,\rho}^{\#,p}(f,g)=d_{\lambda ',\rho}^{\#,p'}(f^*,g^*).\) d) If for every \(\lambda >0\) and \(\rho >0\) \[ \lim_{\epsilon \to 0}d_{\lambda,\rho}^{\#,p}(f_{\epsilon},f)=0, \] then \(f_{\epsilon}\to f\) is the Mosco-epi sense (the converse holds if X is finite dimensional).
scientific article; zbMATH DE number 4050387
scientific article