Outer \(\gamma\)-convexity in normed linear spaces
Let \((X,\|\cdot\|)\) be a normed linear space and \(\gamma\) a fixed positive real number. A subset \(M\subseteq X\) is said to be outer \(\gamma\)-convex if, for all \(x_0\) and \(x_1\) in \(M\), there exist \(k\in\mathbb{N}\) and \(\lambda_i\in [0,1]\), \(i= 0,1,\dots, k\), with \(\lambda_0= 0\), \(\lambda_k= 1\), \(0\leq \lambda_{i+1}- \lambda_i\leq {\gamma\over\|x_0- x_1\|}\) for \(i= 0,1,\dots, k-1\), such that \(x_{\lambda_i}\in M\) for \(i= 0,1,\dots, k\). Given a convex subset \(D\) of \(X\), a function \(f:D\to \mathbb{R}\cup\{\infty\}\) is called outer \(\gamma\)-convex function if, for all \(x_0\) and \(x_1\) in \(D\), then exist \(k\in\mathbb{N}\) and \(\lambda_i\in [0,1]\), \(i= 0,1,\dots, k\), satisfying \(0\leq \lambda_{i+1}- \lambda_i\leq {\gamma\over\|x_0- x_1\|}\) for \(i=0,1,\dots, k-1\) (with \(\lambda_0= 0\), \(\lambda_k=1\)), such that \[ f(x_{\lambda_i})\leq (1- \lambda_i) f(x_0)+ \lambda_i f(x_i)\quad\text{for }0\leq i\leq k. \] In this paper, it is shown that a function \(f: D\subseteq X\to \mathbb{R}\) is outer \(\gamma\)-convex if and only if the level set \(\{x\in D: f(x)+ \xi(x)\leq \alpha\}\) is outer \(\gamma\)-convex for every continuous linear functional \(\xi\in X^*\) and for every real \(\alpha\). Two main properties of outer \(\gamma\)-convex function proved in the paper are: \((M_\gamma)\) Each \(\gamma\)-minimizer (defined by \(f(x^*)\leq f(x)\) for all \(x\in D\) satisfying \(\|x-x^*\|< \gamma+\varepsilon\) for some \(\varepsilon> 0\)) is global minimizer, and \((I_\gamma)\) Each \(\gamma\)-infimizer (defined by \(\liminf_{y\to x^*} f(y)\leq f(x)\) for all \(x\in D\) satisfying \(\|x- x^*\|<\gamma+\varepsilon\) for some \(\varepsilon> 0\)) is a global infimizer (i.e. \(\liminf_{y\to x^*} f(y)\leq f(x)\) for all \(x\in D\)). Other results proved are: (i) Let \(f: [a,b]\to \mathbb{R}\) be lower semicontinuous. Then \(f\) is outer \(\gamma\)-convex if and only if \(f+\xi\) possesses the property \((M_\gamma)\) for every linear functional \(\xi\) on \(\mathbb{R}\). (ii) Let \(f: [a,b]\to \mathbb{R}\) be bounded from below. Then the lower semicontinuous function \(f\) is outer \(\gamma\)-convex if and only if \(f+\xi\) possesses the property \((I_\gamma)\) for every linear functional \(\xi\) on \(\mathbb{R}\). The case \(\gamma=0\) is also discussed in this paper.
scientific article; zbMATH DE number 1485034