Journal of the Mathematical Society of Japan

Convex functions and barycenter on CAT(1)-spaces of small radii


Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We use the convexity of a certain function discovered by W. Kendall on small metric balls in CAT(1)-spaces to show that any probability measure on a complete CAT(1)-space of small radius admits a unique barycenter. We also present various properties of barycenter on those spaces. This extends the results previously known for CAT(0)-spaces and CAT(1)-spaces of small diameter.

Article information

J. Math. Soc. Japan Volume 68, Number 3 (2016), 1297-1323.

First available in Project Euclid: 19 July 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

CAT(1)-space convex function barycenter Banach–Saks property


YOKOTA, Takumi. Convex functions and barycenter on CAT(1)-spaces of small radii. J. Math. Soc. Japan 68 (2016), no. 3, 1297--1323. doi:10.2969/jmsj/06831297.

Export citation


  • B. Afsari, Riemannian $L^p$ center of mass: existence, uniqueness, and convexity, Proc. Amer. Math. Soc., 139 (2011), 655–673.
  • T. Austin, A CAT(0)-valued pointwise ergodic theorem, J. Topol. Anal., 3 (2011), 145–152.
  • M. Bačák, The proximal point algorithm in metric spaces, Israel J. Math., 194 (2013), 689–701.
  • A. Balser and A. Lytchak, Centers of convex subsets of buildings, Ann. Global Anal. Geom., 28 (2005), 201–209.
  • I.-D. Berg and I.-G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 (2008), 195–218.
  • D. Burago, Y. Burago and S. Ivanov, A course in metric geometry, Graduate Studies in Math., 33, Amer. Math. Soc., Providence, RI, 2001.
  • T. Christiansen and K. T. Sturm, Expectations and martingales in metric spaces, Stochastics, 80 (2008), 1–17.
  • I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443–474.
  • R. Espínola and A. Fernández-León, CAT(k)-spaces, weak convergence and fixed points, J. Math. Anal. Appl., 353 (2009), 410–427.
  • A. Es-Sahib and H. Heinich, Barycentre canonique pour un espace métrique courbure négative, (French) Séminaire de Probabilités, XXXIII, 355–370, Lecture Notes in Math., 1709, Springer, Berlin, 1999.
  • K. Fujiwara, K. Nagano and T. Shioya, Fixed point sets of parabolic isometries of CAT(0)-spaces, Comment. Math. Helv., 81 (2006), 305–335.
  • T. Gelander, A. Karlsson and G. A. Margulis, Superrigidity, generalized harmonic maps and uniformly convex spaces, Geom. Funct. Anal., 17 (2008), 1524–1550.
  • T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ., 19 (1979), 215–229.
  • W. Jäger and H. Kaul, Uniqueness and stability of harmonic maps and their Jacobi fields, Manuscripta Math., 28 (1979), 269–291.
  • J. Jost, Equilibrium maps between metric spaces, Calc. Var. Partial Differential Equations, 2 (1994), 173–204.
  • J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Comment. Math. Helv., 70 (1995), 659–673.
  • J. Jost, Generalized harmonic maps between metric spaces, Geometric analysis and the calculus of variations, Int. Press, Cambridge, MA, 1996, 143–174.
  • J. Jost, Y. Xin and L. Yang, The regularity of harmonic maps into spheres and applications to Bernstein problems, J. Differential Geom., 90 (2012), 131–176.
  • S. Kakutani, Weak convergence in uniformly convex spaces, Tôhoku Math. J., 45 (1938), 188–193.
  • H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math., 30 (1977), 509–541.
  • M. Kell, Uniformly Convex Metric Spaces, Anal. Geom. Metr. Spaces, 2 (2014), 359–380.
  • W. Kendall, Probability, convexity, and harmonic maps with small image, I, Uniqueness and fine existence, Proc. London Math. Soc. (3), 61 (1990), 371–406.
  • W. Kendall, Convexity and the hemisphere, J. London Math. Soc. (2), 43 (1991), 567–576.
  • W. Kendall, From stochastic parallel transport to harmonic maps, New directions in Dirichlet forms, 49–115, AMS/IP Stud. Adv. Math., 8, Amer. Math. Soc., Providence, RI, 1998.
  • K. Kuwae, Jensen's inequality over CAT$(\kappa)$-space with small diameter, Potential theory and stochastics in Albac, 173–182, Theta Ser. Adv. Math., 11, Theta, Bucharest, 2009.
  • K. Kuwae, Jensen's inequality on convex spaces, Calc. Var. Partial Differential Equations, 49 (2014), 1359–1378.
  • K. Kuwae and K.-T. Sturm, On a Liouville type theorem for harmonic maps to convex spaces via Markov chains, Proceedings of RIMS Workshop on Stochastic Analysis and Applications, RIMS Kôkyûroku Bessatsu, 2008, 177–191.
  • A. Lytchak, Open map theorem for metric spaces, Algebra i Analiz, 17 (2005), 139–159; translation in St. Petersburg Math. J., 17 (2006), 477–491.
  • A. Navas, An $L^1$ ergodic theorem with values in a non-positively curved space via a canonical barycenter map, Ergodic Theory Dynam. Systems, 33 (2013), 609–623.
  • S.-i. Ohta, Convexities of metric spaces, Geom. Dedicata, 125 (2007), 225–250.
  • S.-i. Ohta, Barycenters in Alexandrov spaces of curvature bounded below, Adv. Geom., 12 (2012), 571–587.
  • T. Sato, An alternative proof of Berg and Nikolaev's characterization of CAT(0)-spaces via quadrilateral inequality, Arch. Math. (Basel), 93 (2009), 487–490.
  • K.-T. Sturm, Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature, Ann. Probab., 30 (2002), 1195–1222.
  • K.-T. Sturm, Probability measures on metric spaces of nonpositive curvature, Heat kernels and analysis on manifolds, graphs, and metric spaces, (Paris, 2002), 357–390, Contemp. Math., 338, AMS, 2003.
  • T. Yokota, A rigidity theorem in Alexandrov spaces with lower curvature bound, Math. Annalen, 353 (2012), 305–331.