The authors consider the quasihyperbolic metric and its geodesics in domains of Banach spaces. The quasihyperbolic metric \(k(x,y)\) was introduced in \(\mathbb{R}^n\) by F. W. Gehring and B. P. Palka [J. Anal. Math. 30, 172–199 (1976; Zbl 0349.30019)] as a substitute for the hyperbolic metric. Three conjectures, concerning local uniqueness and prolongation of geodesics and convexity of quasihyperbolic balls, were posed by J. Väisälä [Ann. Acad. Sci. Fenn., Math. 34, No. 2, 447–473 (2009; Zbl 1186.30026)]. Among these conjectures the convexity conjecture is the strongest. The authors show that if \(\Omega\) is a convex domain in a Banach space, then all quasihyperbolic balls in \(\Omega\) are convex. This generalizes the corresponding result in \(\mathbb{R}^n\) [O. Martio and J. Väisälä, Pure Appl. Math. Q. 7, No. 2, 395–409 (2011; Zbl 1246.30041)]. They also show that if \(\Omega\) is starlike with respect to \(x_0\), then the quasihyperbolic balls centered at \(x_0\) are starlike. Similar results are obtained for the \(j\) metric defined as \(j(x,y) = \log (1+ |x-y|/\min(d(x),d(y))\) where \(d(x)\) denotes the distance from \(x\) to \(\partial \Omega\).