[Banach] Abstract of a paper by Riku Klen, Antti Rasila, and Jarno Talponen

This is an announcement for the paper "On smoothness of quasihyperbolic balls" by Riku Klen, Antti Rasila, and Jarno Talponen.

Abstract: 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 F.W. Gehring and M. 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 renormings of Banach spaces. To provide a useful tool for this approach we turn our attention to the variational stability of quasihyperbolic geodesics. Several examples and illustrations are provided.

Archive classification: math.FA math.CV

Mathematics Subject Classification: 30C65, 46T05, 46B03

Remarks: 19 pages, 4 figures

Submitted from: antti.rasila@iki.fi

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1407.2403

or

http://arXiv.org/abs/1407.2403