Treffer: Semi-Lipschitz Functions and Best Approximation in Quasi-Metric Spaces: Semi-Lipschitz functions and best approximation in quasi-metric spaces

Let \((X,d)\) be a quasi-metric space (symmetry of \(d\) is not satisfied). A function \(f:X\rightarrow \mathbb{R}\) is called semi-Lipschitz if there exists a number \(K\geq 0\) such that \(f\left( x\right) -f\left( y\right) \leq Kd\left( x,y\right) \), for all \(x,y\in X.\) One denotes by \(SLip_0X\) the set of all semi-Lipschitz functions vanishing at a fixed point \(x_0\in X.\) It follows that \(SLip_0X\) is a semilinear space and the functional \(\left\|\cdot \right\|_d\) defined by \(\left\|f\right\|_d=\sup \left\{ ((f\left( x\right) -f\left( y\right))\vee 0)/d\left( x,y\right) \right. :\left. x,y\in X,d\left( x,y\right) >0\right\}\) is a quasi-norm on \(SLip_0X.\) For a subset \(Y\) of \(X\) containing \(x_0\) and \(p\in X\) let \( P_Y(p)=\{y_0\in Y:d(y_0,p)\) \(=\inf \{d(y,p):y\in Y\}\}.\) The authors give characterizations of the elements of \(P_Y\left( p\right) \) in terms of the elements of \(SLip_0X.\) One obtains results similar to those obtained in the case of metric spaces and the spaces of Lipschitz functions on them (which in their turn are inspired by the characterizations of the elements of best approximation in normed spaces in terms of the elements of their duals). The completeness of the space \(SLip_0X\) is also proved. As the authors point out in the introduction, other properties of \(SLip_0X\) (compactness, the property of being a Banach space etc.) will be studied elsewhere.