On the smoothness of quasihyperbolic balls

Riku Klén; Antti Rasila; Jarno Talponen

doi:10.5186/aasfm.2017.4226

On the smoothness of quasihyperbolic balls

Riku Klén, Antti Rasila, Jarno Talponen

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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 C¹-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.

Original language	English
Pages (from-to)	439-452
Number of pages	14
Journal	Annales Academiae Scientiarum Fennicae Mathematica
Volume	42
Issue number	1
DOIs	https://doi.org/10.5186/aasfm.2017.4226
State	Published - 2017
Externally published	Yes

Keywords

Convexity
Geodesics
Quasihyperbolic metric
Renorming
Smoothness
Uniqueness

Access to Document

10.5186/aasfm.2017.4226

Cite this

@article{ac2ee41d89ee4bc89bb314f51806724f,

title = "On the smoothness of quasihyperbolic balls",

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 C1-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.",

keywords = "Convexity, Geodesics, Quasihyperbolic metric, Renorming, Smoothness, Uniqueness",

author = "Riku Kl{\'e}n and Antti Rasila and Jarno Talponen",

year = "2017",

doi = "10.5186/aasfm.2017.4226",

language = "英语",

volume = "42",

pages = "439--452",

journal = "Annales Academiae Scientiarum Fennicae Mathematica",

issn = "1239-629X",

publisher = "Finnish Mathematical Society",

number = "1",

}

TY - JOUR

T1 - On the smoothness of quasihyperbolic balls

AU - Klén, Riku

AU - Rasila, Antti

AU - Talponen, Jarno

PY - 2017

Y1 - 2017

N2 - 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 C1-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.

AB - 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 C1-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.

KW - Convexity

KW - Geodesics

KW - Quasihyperbolic metric

KW - Renorming

KW - Smoothness

KW - Uniqueness

UR - http://www.scopus.com/inward/record.url?scp=85012154153&partnerID=8YFLogxK

U2 - 10.5186/aasfm.2017.4226

DO - 10.5186/aasfm.2017.4226

M3 - 文章

AN - SCOPUS:85012154153

SN - 1239-629X

VL - 42

SP - 439

EP - 452

JO - Annales Academiae Scientiarum Fennicae Mathematica

JF - Annales Academiae Scientiarum Fennicae Mathematica

IS - 1

ER -

On the smoothness of quasihyperbolic balls

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this