Conical geodesic bicombings on subsets of normed vector spaces
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Giuliano Basso
und
Benjamin Miesch
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.
Communicated by: M. Henk
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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© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- On the cells in a stationary Poisson hyperplane mosaic
- Conical geodesic bicombings on subsets of normed vector spaces
- Variance estimates and almost Euclidean structure
- Special cubic Cremona transformations of ℙ6 and ℙ7
- Lipschitz–Killing curvatures and polar images
- On the connectivity of the hyperbolicity region of irreducible polynomials
- Biharmonic hypersurfaces in 5-dimensional non-flat space forms
- On the principal Ricci curvatures of a Riemannian 3-manifold
- On the curve Yn = Xℓ(Xm + 1) over finite fields
- Geometries arising from trilinear forms on low-dimensional vector spaces
Artikel in diesem Heft
- Frontmatter
- On the cells in a stationary Poisson hyperplane mosaic
- Conical geodesic bicombings on subsets of normed vector spaces
- Variance estimates and almost Euclidean structure
- Special cubic Cremona transformations of ℙ6 and ℙ7
- Lipschitz–Killing curvatures and polar images
- On the connectivity of the hyperbolicity region of irreducible polynomials
- Biharmonic hypersurfaces in 5-dimensional non-flat space forms
- On the principal Ricci curvatures of a Riemannian 3-manifold
- On the curve Yn = Xℓ(Xm + 1) over finite fields
- Geometries arising from trilinear forms on low-dimensional vector spaces
