Geometries induced by logarithmic oscillations as examples of Gromov hyperbolic spaces (Q523217)

A metric space \((M,d)\) is called \textit{Gromov hyperbolic} if there exists some constant \(\delta \geq 0\) such that \((x|y)_w \geq (x|z)_w \land (z|y)_w-\delta\), for \(x,y,z,w\in M\), where \((x|y)_z=(1/2)[d(x,z)+d(y,z)-d(x,y)] \) and \(a\land b=\min \{a,b\}\). \((x|y)_z\) is called the \textit{Gromov product} of \(x\) and \(y\) with respect to \(z\), The authors start with simple metric spaces, and include some results that were proved in other papers but with different methods. In Section 2 (Fundamental exploration), they show that the metric space \((\mathbb{R}^n ,d)\), with \(d(x,y):=\ln (1+|x-y|)\), is Gromov hyperbolic (Theorem 2.1). In Section 3 (Classical geometries and Gromov hyperbolicity), they show that the Euclidean space \((\mathbb{R}^2, |\cdot |)\) is not Gromov hyperbolic (Theorem 3.1) but the Euclidean distance on the infinite strip \(\{(x,y)\in \mathbb{R}^2:0<y<1\}\) is (Theorem 3.2). Let us now consider the unit disk \(D\) centered at the origin and \(X\) and \(Y\) two points in the interior of \(D\). Denote by \(s\), \(S\) the intersection points of the circle \(\partial D\) with the line \(XY\), such that the order of the points in the line is \(s,X,Y,S\). The Hilbert distance between \(X\) and \(y\) is given by \[ h(X,Y):=\ln\frac {SX}{SY}\cdot\frac{sY}{sX} \] (where \(SX\)\dots are the Euclidean distances between the points). The unit disk \(D\) endowed with Hilbert's distance is Gromov hyperbolic (Theorem 3.3). Let \(K\) be a circle centered at the origin and let \(s,X,Y,S\) be as above. The half Hilbert distance in the interior of the disk designated by \(J\) in terms of anharmonic ratio is defined by \(d^H(X,Y)=\frac{1}{2} \ln [XYSs]\). Let \(g:= Ss\) be the arc of circle orthogonal to \(K\) and let \(F_1=OX\bigcap g\) and \(F_2=OY\bigcap g\). Let \([F_1F_2Ss]_g\) be the anharmonic ratio on the orthogonal \(g\). Let \(d^P(F_1,F_2):=[F_1F_2Ss]_g\) be the Poincaré distance and let \[ d^*(F_1,F_")=\ln\frac{\max_{P\in K}\frac{PF_2}{PF_1}}{\min_{P\in K}\frac{PF_2}{PF_1}} \] be the Barbilian distance. The Barbilian distance, the Poincaré distance, and the half Hilbert distance coincide on the unit disk \(D\) (Theorem 3.5). The Barbilian distance and the Poincaré distance are Gromov hyperbolic on the disk (Theorem 3.6) and Poincaré's half plane \(\mathbb{H}\) endowed with the Poincaré distance is Gromov hyperbolic. This section ends with the proof that the sphere of radius 1/2 endowed with the chordal distance \(q(x,y):=|\pi (x)-\pi (y)|\) where \(\pi (x) \) and \(\pi (y)\) are the points on the sphere that stereographically project on \(x\) and \(y\), is not Gromov hyperbolic. In Section 4, (Barbilian's logarithmic oscillation and the stabilizing distance) the authors ``bridge the gap between the developments in the study of geometries generated by logarithmic oscillation and the study of Gromov hyperbolic spaces''. They prove that if \(G=\mathbb{R}\setminus \{0\}\) then \(d_B(x,y)=\ln\frac {\delta _x \lor \delta _y}{\delta _x \land \delta _y} \) where \(\delta_x\) is the Euclidean distance between \(x\) and the origin \(0\), is a semi-distance in \(G\) and endowed with \(d_B\), \(G\) is Gromov hyperbolic with \(\delta =0\) (Theorem 4.1) and \(a\lor b=\max\{a,b\}\). \(d_B\) is actually half the Barbilian's classical distance. Let \(G\) be an arbitrary subset in \(\mathbb{R}^n\). For any \(x\) and \(y\) in \( \mathbb{R}^n\) let \(|x-y|\) be the Euclidean distance between \(x\) and \(y\). Let \(M\in \mathbb{R}^n\) be such that \(M\bigcap G=\emptyset\). Let \(\delta _x= \min_{z\in M}|x-z|\) with \(x\in G\). Let \[ d_{G,M}(x,y)=\ln\left( \frac { \delta _x \lor \delta _y}{\delta _x \land \delta _y} + |x-y|\right). \] \(d_{G,M}\) is a distance in \(G\) and if \(G\) is the punctured ball and \(M={0}\) then this metric is Gromov hyperbolic with \(\delta =\frac{1}{2} \ln 9\) (Theorem 4.2). In Section 5 (Extensions of the stabilizing metric), the authors study distances \(d_f(x,y)=\left( \frac { \delta _x \lor \delta _y}{\delta _x \land \delta _y} + f(|x-y|)\right)\) where \(f\) is a function satisfying predetermined restrictions. In the last section (Extensions of Vuorinen's metric), the authors do a similar construction as before but they use the metric \(j_G(x,y)=\ln\left( 1+\frac{|x-y|}{\delta _x \land \delta _y}\right) \) and analyse the stabilizing metric \[ d_{VH,s}=\ln \left(\frac { \delta _x \lor \delta _y}{\delta _x \land \delta _y} + \frac{|x-y|}{\delta _x \land \delta _y}\right). \]