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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

IMPACT FACTOR 2018: 0.789

CiteScore 2018: 0.73

SCImago Journal Rank (SJR) 2018: 0.329
Source Normalized Impact per Paper (SNIP) 2018: 0.908

Mathematical Citation Quotient (MCQ) 2018: 0.53

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Volume 19, Issue 2


Conical geodesic bicombings on subsets of normed vector spaces

Giuliano Basso / Benjamin Miesch
Published Online: 2019-04-09 | DOI: https://doi.org/10.1515/advgeom-2018-0008


We establish existence and uniqueness results for conical geodesic bicombings on subsets of normed vector spaces. Concerning existence, we give a first example of a convex geodesic bicombing that is not consistent. Furthermore, we show that under a mild geometric assumption on the norm a conical geodesic bicombing on an open subset of a normed vector space locally consists of linear geodesics. As an application, we obtain by the use of a Cartan–Hadamard type result that if a closed convex subset of a Banach space has non-empty interior, then it admits a unique consistent conical geodesic bicombing, namely the one given by the linear segments.

Keywords: Nonpositive curvature; geodesic bicombing; convex sets

MSC 2010: 46B20; 46B22; 51F99; 53C22


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About the article

Received: 2016-09-02

Revised: 2017-03-29

Published Online: 2019-04-09

Published in Print: 2019-04-24

Communicated by: M. Henk

Funding The authors gratefully acknowledge support from the Swiss National Science Foundation.

Citation Information: Advances in Geometry, Volume 19, Issue 2, Pages 151–164, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0008.

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