Weak convexity of functions and the infimal convolution (Q2821971)

Let \(E\) be a real Banach space. The infimal convolution of two functions \(f: E\to\overline{\mathbb{R}}=\mathbb{R}\cup\{-\infty,\infty\}\) and \(g:E\to\overline{\mathbb{R}}\) is the function \(f\boxplus g\), defined by NEWLINE\[NEWLINE(f\boxplus g)(x)= \text{inf}_{x_1,x_2\in E:x_1+x_2= x}(f(x_1)+ g(x_2)),\;x\in E.NEWLINE\]NEWLINE The sup-convolution of \(f\) and \(g\) is defined as NEWLINE\[NEWLINE(f\boxminus g)(x)= \sup_{x_1,x_2\in E:x_1+x_2= x} (f(x_1)+ (-g(x_2))),\;x\in E.NEWLINE\]NEWLINE For a family \(\Phi\) of functions \(\varphi: E\to\mathbb{R}\), a function \(f:E\to\overline{\mathbb{R}}\) is said to be \(\Phi\)-convex if there exists a subset \(\Phi'\subset \Phi\) such that NEWLINE\[NEWLINEf(x)= \sup_{\varphi\in\Phi'}\varphi(x),\;x\in E.NEWLINE\]NEWLINE NEWLINEA function \(f:E\to \mathbb{R}\cup\{+\infty\}\) is said to be convolutionally weakly convex with respect to (w.r.t.) \(\gamma:E\to\mathbb{R}\) if it is \(\Phi\)-convex with \(\Phi=\Phi(\gamma)\). The class of such functions is denoted by \({\mathcal C}{\mathcal W}{\mathcal C}(\gamma)\).NEWLINENEWLINE NEWLINEGiven a function \(\gamma:E\to \overline{\mathbb{R}}\) and a number \(t>0\), we denote by \(\gamma_t(x)= t\), \(\gamma({x\over t})\), \(x\in E\). A function \(f\in{\mathcal C}{\mathcal W}{\mathcal C}(\gamma)\) is called regularly convolutionally weakly convolutionally weakly convex w.r.t. \(\gamma\) if NEWLINE\[NEWLINE\forall\varepsilon\in (0,1)\,\exists\delta\in (0,1): \forall t\in (0,\delta)\, f\boxplus\gamma_t\in {\mathcal C}{\mathcal W}{\mathcal C}(\gamma_{1-\varepsilon}).NEWLINE\]NEWLINE NEWLINEThe class of such functions is denoted by \({\mathcal C}{\mathcal W}{\mathcal C}(\gamma)\). The \(\gamma\)-predifferential of a function \(f:E\to\overline{\mathbb{R}}\) at a pint \(x\in\text{dom\,}f\) is defined by NEWLINE\[NEWLINE\pi_\gamma f(x)= \{u\in \text{dom\,}\gamma\mid\exists t> 0:(f\boxplus\gamma_t) (x+tu)= f(x)+ \gamma_t(tu)\}.NEWLINE\]NEWLINE A function \(f: E\to\mathbb{R}\cup\{+\infty\}\) is said to be weakly convex w.r.t. \(\gamma:E\to\mathbb{R}\) if NEWLINE\[NEWLINE(f\boxplus\gamma) (x+ u)= f(x)+ \gamma(u),\;\forall x\in\text{dom\,}f,\;\forall u\in \pi_\gamma f(x).NEWLINE\]NEWLINE We denote by \({\mathcal W}{\mathcal C}(\gamma)\) the class of weakly convex (w.r.t. \(\gamma\)) lower semicontinuous functions \(f: E\to\mathbb{R}\cup\{+\infty\}\) such that \(\text{dom}(f\boxplus\gamma)\neq\varphi\).NEWLINENEWLINE In terms of sup-inf convolutions, the author characterizes the class \({\mathcal C}{\mathcal W}{\mathcal C}(\gamma)\). Motivated by Moreau-Yosida regularization \((f\boxplus B_t)\) and Lasry-Lions regularization \(f\boxminus \beta_t\boxplus\beta_s\), where \(\beta_t(x)= {1\over 2}\| x\|^2\), \(t>0\), \(x\in E\). He considers a subclass of \(\text{CWE}(\gamma)\), namely, the class \({\mathcal R}{\mathcal C}{\mathcal W}{\mathcal C}(\gamma)\) and shows that under reasonable assumptions on \(\gamma\) the class \({\mathcal R}{\mathcal C}{\mathcal W}{\mathcal C}(\gamma)\) is the same as the class \({\mathcal W}{\mathcal C}(\gamma)\).