Convexity properties of quasihyperbolic balls on Banach spaces (Q2906162)

\textit{R. Klén} has studied in [J. Math. Anal. Appl. 342, No. 1, 192--201 (2008; Zbl 1166.30023); Ann. Acad. Sci. Fenn., Math. 33, No. 1, 281--293 (2008; Zbl 1149.30035)] the convexity of metric balls defined by the distance ratio or the quasihyperbolic metric of a domain in \(\mathbb{R}^n\). The authors generalise these results to the Banach space setting. For a domain \(\Omega \subsetneq X\) of a Banach space \(X\) let \(d(x)= d(x, \partial \Omega)\) and NEWLINE\[NEWLINE j(x,y) = \log \left( 1+ \frac{|x-y|}{\min\{d(x), d(y)\}}\right) NEWLINE\]NEWLINE be the distance ratio metric and let \(k(x,y)\) be the quasihyperbolic metric of \( \Omega\). The authors prove that each \(j\)-ball NEWLINE\[NEWLINE B_j(x_0,r) = \{ x \in \Omega : j(x_0,x)<r \} NEWLINE\]NEWLINE is starlike with respect to \(x_0\) for all \(r \leq \log 2\). Moreover, they prove that if \(\Omega \subsetneq X\) is a convex domain of \(X\), then all \(j\)-balls and quasihyperbolic balls are convex.NEWLINENEWLINE\textit{O. Martio} and \textit{J. Väisälä} [Pure Appl. Math. Q. 7, No. 2, 395--409 (2011; Zbl 1246.30041)] have posed the question whether the hyperbolic balls of convex domains of uniformly convex Banach spaces are quasihyperbolically convex. The authors give a negative answer to this question.