Close-to-convexity of quasihyperbolic and jj-metric balls

We will consider close-to-convexity of the metric balls defined by the quasihyperbolic metric and the $j$-metric. We will show that the $j$-metric balls with small radii are close-to-convex in general subdomains of \Rn and the quasihyperbolic balls with small radii are close-to-convex in the punctured space.Comment: 12 pages, 2 figure

Dimension of the boundary in different metrics

We consider metrics on Euclidean domains $\Omega\subset\R^n$ that are induced by continuous densities $\rho\colon\Omega\rightarrow(0,\infty)$ and study the Hausdorff and packing dimensions of the boundary of $\Omega$ with respect to these metrics.Comment: 20 pages, 2 figure