We investigate properties of quasihyperbolic balls and geodesics in Euclidean and Banach spaces. Our main result is that in uniformly smooth Banach spaces a quasihyperbolic ball of a convex domain is C-1-smooth. The question about the smoothness of quasihyperbolic balls is old, originating back to the discussions of Gehring and Vuorinen in 1970's. To our belief, the result is new also in the Euclidean setting. We also address some other issues involving the smoothness of quasihyperbolic balls. We introduce an interesting application of quasihyperbolic metrics to equivalent renormings of Banach spaces. Several examples and illustrations are provided.
|Number of pages||14|
|Journal||ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA|
|Publication status||Published - 2017|
|MoE publication type||A1 Journal article-refereed|
- Quasihyperbolic metric
- CONVEXITY PROPERTIES
- UNIFORM DOMAINS