A common generalization of the postman, radial, and river metrics (Q2871491)

Let \((X,d)\) be a metric space, \(\varphi\) be a function from \(X\) to \(X\), \(\rho\) be a metric on the range \(\varphi(X)\) of \(\varphi\) and let \(\Gamma\) be a relation on \(X^2\) to \(X\). The author defines a generalized equivalence relation \(Q_{\varphi\Gamma}\) on \(X\) and a generalized metric \(d_{\rho\varphi}(x,y)\) on \(X\) as NEWLINE\[NEWLINE Q_{\varphi\Gamma}=\{(x,y)\in X^2:\varphi(x)=\varphi(y)\in\Gamma(x,y)\} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE d_{\rho\varphi}(x,y)=d(x,\varphi(x))+\rho(\varphi(x),\varphi(y))+d(\varphi(y),y). NEWLINE\]NEWLINE Let NEWLINE\[NEWLINE d_{\rho\varphi\Gamma}(x,y)=\begin{cases} d(x,y)\,\, &\text{if }(x,y)\in Q_{\varphi\Gamma},\\ d_{\rho\varphi}(x,y)\,\, &\text{if }(x,y)\not\in Q_{\varphi\Gamma}. \end{cases} NEWLINE\]NEWLINE It is proved that \(d_{\rho\varphi\Gamma}\) is a metric on \(X\) which includes the well-known postman, radial and river metrics as particular cases. Some properties of \(d_{\rho\varphi\Gamma}\) are analyzed. For example, it is proved that the inequality \( d(x,y)\leq d_{\rho\varphi Q}(x,y) \) holds for all \(x,y\in X\) whenever \(d(u,v)\leq\rho(u,v)\) for all \(u,v\in\varphi[X]\).NEWLINENEWLINEFurthermore the author introduces and studies \(\varphi\)-metrics on \(X\) and collinearity-like relations on \(X^2\).NEWLINENEWLINEA relation \(\Gamma\) on \(X^2\) to \(X\) is a collinearity relation on \(X\) if, for all \(x,y,z\in X\), \(\Gamma(x,x)=x\), \(\Gamma(x,y)=\Gamma(y,x)\), \(\Gamma(x,y)\cap\Gamma(y,z)\cap\{y\}^c\subseteq\Gamma(x,z)\) and \( (z\in\Gamma(x,y))\Rightarrow(x\in\Gamma(y,z)). \)NEWLINENEWLINEThe author shows that if \(X\) is a vector space over \(K\) and NEWLINE\[NEWLINE \Gamma(x,y)=\begin{cases} X\,\, &\text{if }x=y,\\ \{z\in X:\exists\lambda\in K: z=\lambda x+(1-\lambda)y\}\,\, &\text{if }x\neq y, \end{cases} NEWLINE\]NEWLINE then \(\Gamma\) is a collinearity relation. Thus the above defined collinearity relation is a natural generalization of the usual collinearity.NEWLINENEWLINEA symmetric function \(d:X^2\to\mathbb R^+\) is called a \(\varphi\)-metric if the following conditions hold for all \(x,y,z\in X\): \(d(x,y)\leq d(x,,y)+d(y,z)-d(y,y)\) and \( (d(x,y)=0)\Leftrightarrow(x=y=\varphi(x)=\varphi(y)). \)NEWLINENEWLINENote that every \(\varphi\)-metric is a weak partial pseudo-metric by \textit{R. Heckmann} [Appl. Categ. Struct. 7, No. 1--2, 71--83 (1999; Zbl 0993.54029)].