Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter April 9, 2019

Conical geodesic bicombings on subsets of normed vector spaces

  • Giuliano Basso EMAIL logo and Benjamin Miesch
From the journal Advances in Geometry


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.


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.


[1] C. D. Aliprantis, K. C. Border, Infinite dimensional analysis. Springer 2006. MR2378491 Zbl 1156.46001Search in Google Scholar

[2] J. M. Alonso, M. R. Bridson, Semihyperbolic groups. Proc. London Math. Soc. (3)70 (1995), 56–114. MR1300841 Zbl 0823.2003510.1112/plms/s3-70.1.56Search in Google Scholar

[3] G. Basso, Fixed point theorems for metric spaces with a conical geodesic bicombing. Ergodic Theory and Dynamical Systems38 (2018), 1642–1657. MR381999610.1017/etds.2016.106Search in Google Scholar

[4] H. Busemann, B. B. Phadke, Spaces with distinguished geodesics. Dekker 1987. MR896903 Zbl 0631.53001Search in Google Scholar

[5] D. Descombes, Asymptotic rank of spaces with bicombings. Math. Z. 284 (2016), 947–960. MR3563261 Zbl 1360.5307810.1007/s00209-016-1680-3Search in Google Scholar

[6] D. Descombes, U. Lang, Convex geodesic bicombings and hyperbolicity. Geom. Dedicata177 (2015), 367–384. MR3370039 Zbl 1343.5303610.1007/s10711-014-9994-ySearch in Google Scholar

[7] D. Descombes, U. Lang, Flats in spaces with convex geodesic bicombings. Anal. Geom. Metr. Spaces4 (2016), 68–84. MR3483604 Zbl 1341.5307010.1515/agms-2016-0003Search in Google Scholar

[8] A. W. M. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces. Adv. in Math. 53 (1984), 321–402. MR753872 Zbl 0562.5404110.1016/0001-8708(84)90029-XSearch in Google Scholar

[9] S. Gähler, G. Murphy, A metric characterization of normed linear spaces. Math. Nachr. 102 (1981), 297–309. MR642160 Zbl 0499.4600710.1002/mana.19811020125Search in Google Scholar

[10] D. B. Goodner, Projections in normed linear spaces. Trans. Amer. Math. Soc. 69 (1950), 89–108. MR0037465 Zbl 0041.2320310.1090/S0002-9947-1950-0037465-6Search in Google Scholar

[11] J. R. Isbell, Six theorems about injective metric spaces. Comment. Math. Helv. 39 (1964), 65–76. MR0182949 Zbl 0151.3020510.1007/BF02566944Search in Google Scholar

[12] M. Kell, Sectional curvature-type conditions on finsler-like metric spaces. Preprint 2016, arXiv:1601.03363 [math.MG]Search in Google Scholar

[13] J. L. Kelley, Banach spaces with the extension property. Trans. Amer. Math. Soc. 72 (1952), 323–326. MR0045940 Zbl 0046.1200210.1090/S0002-9947-1952-0045940-5Search in Google Scholar

[14] U. Lang, Injective hulls of certain discrete metric spaces and groups. J. Topol. Anal. 5 (2013), 297–331. MR3096307 Zbl 1292.2004610.1142/S1793525313500118Search in Google Scholar

[15] Y.-C. Li, C.-C. Yeh, Some characterizations of convex functions. Comput. Math. Appl. 59 (2010), 327–337. MR2575519 Zbl 1189.2601310.1016/j.camwa.2009.05.020Search in Google Scholar

[16] B. Miesch, The Cartan–Hadamard Theorem for Metric Spaces with Local Geodesic Bicombings. Enseign. Math. 63 (2017) 231–245 MR3777137 Zbl 1391.5308810.4171/LEM/63-1/2-8Search in Google Scholar

[17] S. Reich, I. Shafrir, Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15 (1990), 537–558. MR1072312 Zbl 0728.4704310.1016/0362-546X(90)90058-OSearch in Google Scholar

[18] W. Takahashi, A convexity in metric space and nonexpansive mappings. I. Kōdai Math. Sem. Rep. 22 (1970), 142–149. MR0267565 Zbl 0268.5404810.2996/kmj/1138846111Search in Google Scholar

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

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 1.6.2023 from
Scroll to top button