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Conical geodesic bicombings on subsets of normed vector spaces

Giuliano Basso and Benjamin Miesch
From the journal Advances in Geometry

Abstract

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.

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

  1. Communicated by: M. Henk

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

Acknowledgement

We would like to thank Urs Lang for introducing us to conical geodesic bicombings and for his helpful remarks and guidance. We are also thankful for helpful suggestions of the anonymous referee.

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Received: 2016-09-02
Revised: 2017-03-29
Published Online: 2019-04-09
Published in Print: 2019-04-24

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