Close-to-convexity of quasihyperbolic and $j$-metric balls

Abstract

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