SpringerLink
Log in

Old and new challenges in Hadamard spaces

  • Published:
Japanese Journal of Mathematics Aims and scope

Abstract

Hadamard spaces have traditionally played important roles in geometry and geometric group theory. More recently, they have additionally turned out to be a suitable framework for convex analysis, optimization and non-linear probability theory. The attractiveness of these emerging subject fields stems, inter alia, from the fact that some of the new results have already found their applications both in mathematics and outside. Most remarkably, a gradient flow theorem in Hadamard spaces was used to attack a conjecture of Donaldson in Kähler geometry. Other areas of applications include metric geometry and minimization of submodular functions on modular lattices. There have been also applications into computational phylogenetics and image processing.

We survey recent developments in Hadamard space analysis and optimization with the intention to advertise various open problems in the area. We also point out several fallacies in the existing proofs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

References

  1. A.D. Aleksandrov, A theorem on triangles in a metric space and some of its applications, Trudy Mat. Inst. Steklov., 38 (1951), 5–23.

    MathSciNet  Google Scholar 

  2. S. Alexander, V. Kapovitch and A. Petrunin, An Invitation to Alexandrov Geometry: CAT(0) Spaces, SpringerBriefs Math., Springer-Verlag, 2019.

  3. L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures. Second ed., Lectures Math. ETH Zürich, Birkhäuser Verlag, Basel, 2008.

    MATH  Google Scholar 

  4. L. Ambrosio and B. Kirchheim, Currents in metric spaces, Acta Math., 185 (2000), 1–80.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Andoni, A. Naor and O. Neiman, Snowflake universality of Wasserstein spaces, Ann. Sci. Éc. Norm. Supér. (4), 51 (2018), 657–700.

    Article  MathSciNet  MATH  Google Scholar 

  6. F. Ardila, T. Baker and R. Yatchak, Moving robots efficiently using the combinatorics of CAT(0) cubical complexes, SIAM J. Discrete Math., 28 (2014), 986–1007.

    Article  MathSciNet  MATH  Google Scholar 

  7. F. Ardila, H. Bastidas, C. Ceballos and J. Guo, The configuration space of a robotic arm in a tunnel, SIAM J. Discrete Math., 31 (2017), 2675–2702.

    Article  MathSciNet  MATH  Google Scholar 

  8. F. Ardila, M. Owen and S. Sullivant, Geodesics in CAT(0) cubical complexes, Adv. in Appl. Math., 48 (2012), 142–163.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Ariza-Ruiz, G. López-Acedo and A. Nicolae, The asymptotic behavior of the composition of firmly nonexpansive map**s, J. Optim. Theory Appl., 167 (2015), 409–429.

    Article  MathSciNet  MATH  Google Scholar 

  10. Z. Artstein and R.J.-B. Wets, Consistency of minimizers and the SLLN for stochastic programs, J. Convex Anal., 2 (1995), 1–17.

    MathSciNet  MATH  Google Scholar 

  11. H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman, Advanced Publ. Program, Boston, MA, 1984.

    MATH  Google Scholar 

  12. T. Austin, A CAT(0)-valued pointwise ergodic theorem, J. Topol. Anal., 3 (2011), 145–152.

    Article  MathSciNet  MATH  Google Scholar 

  13. T. Austin, Erratum: A CAT(0)-valued pointwise ergodic theorem, J. Topol. Anal., 8 (2016), 373–374.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Bačák, The proximal point algorithm in metric spaces, Israel J. Math., 194 (2013), 689–701.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Bačák, Computing medians and means in Hadamard spaces, SIAM J. Optim., 24 (2014), 1542–1566.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Bačák, Convex Analysis and Optimization in Hadamard Spaces, De Gruyter Ser. Nonlinear Anal. Appl., 22, De Gruyter, Berlin, 2014.

    Book  MATH  Google Scholar 

  17. M. Bačák, A new proof of the Lie–Trotter–Kato formula in Hadamard spaces, Commun. Contemp. Math., 16 (2014), no. 6, 1350044.

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Bačák, Convergence of nonlinear semigroups under nonpositive curvature, Trans. Amer. Math. Soc., 367 (2015), 3929–3953.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Bačák, A variational approach to stochastic minimization of convex functionals, Pure Appl. Funct. Anal., 3 (2018), 287–295.

    MathSciNet  MATH  Google Scholar 

  20. M. Bačák, R. Bergmann, G. Steidl and A. Weinmann, A second order non-smooth variational model for restoring manifold-valued images, SIAM J. Sci. Comput., 38 (2016), A567–A597.

    Article  MATH  Google Scholar 

  21. M. Bačák and L.V. Kovalev, Lipschitz retractions in Hadamard spaces via gradient flow semigroups, Canad. Math. Bull., 59 (2016), 673–681.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Bačák, M. Montag and G. Steidl, Convergence of functions and their Moreau envelopes on Hadamard spaces, J. Approx. Theory, 224 (2017), 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Bačák and S. Reich, The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces, J. Fixed Point Theory Appl., 16 (2014), 189–202.

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Bačák, I. Searston and B. Sims, Alternating projections in CAT(0) spaces, J. Math. Anal. Appl., 385 (2012), 599–607.

    Article  MathSciNet  MATH  Google Scholar 

  25. J.-B. Baillon, Un exemple concernant le comportement asymptotique de la solution du problème du/dt+∂φ(u) ∋ 0, J. Functional Analysis, 28 (1978), 369–376.

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Banert, Backward-backward splitting in Hadamard spaces, J. Math. Anal. Appl., 414 (2014), 656–665.

    Article  MathSciNet  MATH  Google Scholar 

  27. D. Barden and H. Le, The logarithm map, its limits and Fréchet means in orthant spaces, Proc. Lond. Math. Soc. (3), 117 (2018), 751–789.

    Article  MathSciNet  MATH  Google Scholar 

  28. D. Barden, H. Le and M. Owen, Central limit theorems for Fréchet means in the space of phylogenetic trees, Electron. J. Probab., 18 (2013), no. 25.

  29. D. Barden, H. Le and M. Owen, Limiting behaviour of Fréchet means in the space of phylogenetic trees, Ann. Inst. Statist. Math., 70 (2018), 99–129.

    Article  MathSciNet  MATH  Google Scholar 

  30. H.H. Bauschke, The composition of projections onto closed convex sets in Hilbert space is asymptotically regular, Proc. Amer. Math. Soc., 131 (2003), 141–146.

    Article  MathSciNet  MATH  Google Scholar 

  31. H.H. Bauschke, J.V. Burke, F.R. Deutsch, H.S. Hundal and J.D. Vanderwerff, A new proximal point iteration that converges weakly but not in norm, Proc. Amer. Math. Soc., 133 (2005), 1829–1835.

    Article  MathSciNet  MATH  Google Scholar 

  32. H.H. Bauschke and P.L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces. Second ed., With a foreword by Hédy Attouch, CMS Books Math./Ouvrages Math. SMC, Springer-Verlag, 2017.

  33. H.H. Bauschke, E. Matoušková and S. Reich, Projection and proximal point methods: convergence results and counterexamples, Nonlinear Anal., 56 (2004), 715–738.

    Article  MathSciNet  MATH  Google Scholar 

  34. P. Benner, M. Bačák and P.-Y. Bourguignon, Point estimates in phylogenetic reconstructions, Bioinformatics, 30 (2014), i534–i540.

    Article  Google Scholar 

  35. A. Bërdëllima, A note on Mosco convergence in CAT(0) spaces, Canad. Math. Bull., 65 (2022), 994–1003.

    Article  MathSciNet  MATH  Google Scholar 

  36. I.D. Berg and I.G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 (2008), 195–218.

    Article  MathSciNet  MATH  Google Scholar 

  37. R. Bergmann, R.H. Chan, R. Hielscher, J. Persch and G. Steidl, Restoration of manifold-valued images by half-quadratic minimization, Inverse Probl. Imaging, 10 (2016), 281–304.

    Article  MathSciNet  MATH  Google Scholar 

  38. R. Bergmann, F. Laus, G. Steidl and A. Weinmann, Second order differences of cyclic data and applications in variational denoising, SIAM J. Imaging Sci., 7 (2014), 2916–2953.

    Article  MathSciNet  MATH  Google Scholar 

  39. R. Bergmann, J. Persch and G. Steidl, A parallel Douglas–Rachford algorithm for minimizing ROF-like functionals on images with values in symmetric Hadamard manifolds, SIAM J. Imaging Sci., 9 (2016), 901–937.

    Article  MathSciNet  MATH  Google Scholar 

  40. R. Bergmann and A. Weinmann, Inpainting of cyclic data using first and second order differences, preprint, ar**v:1410.1998v1; Accepted converence paper at EMMCVPR’15.

  41. R. Bergmann and A. Weinmann, A second-order TV-type approach for inpainting and denoising higher dimensional combined cyclic and vector space data, J. Math. Imaging Vision, 55 (2016), 401–427.

    Article  MathSciNet  MATH  Google Scholar 

  42. R.J. Berman, T. Darvas and C.H. Lu, Convexity of the extended K-energy and the large time behavior of the weak Calabi flow, Geom. Topol., 21 (2017), 2945–2988.

    Article  MathSciNet  MATH  Google Scholar 

  43. D.P. Bertsekas, Incremental proximal methods for large scale convex optimization, Math. Program., 129 (2011), 163–195.

    Article  MathSciNet  MATH  Google Scholar 

  44. L.J. Billera, S.P. Holmes and K. Vogtmann, Geometry of the space of phylogenetic trees, Adv. in Appl. Math., 27 (2001), 733–767.

    Article  MathSciNet  MATH  Google Scholar 

  45. K. Borsuk and S. Ulam, On symmetric products of topological spaces, Bull. Amer. Math. Soc., 37 (1931), 875–882.

    Article  MathSciNet  MATH  Google Scholar 

  46. J. Bourgain, On Lipschitz embedding of finite metric spaces in Hilbert space, Israel J. Math., 52 (1985), 46–52.

    Article  MathSciNet  MATH  Google Scholar 

  47. J. Bourgain, The metrical interpretation of superreflexivity in Banach spaces, Israel J. Math., 56 (1986), 222–230.

    Article  MathSciNet  MATH  Google Scholar 

  48. J. Bourgain, V. Milman and H. Wolfson, On type of metric spaces, Trans. Amer. Math. Soc., 294 (1986), 295–317.

    Article  MathSciNet  MATH  Google Scholar 

  49. T. Brady and J. McCammond, Braids, posets and orthoschemes, Algebr. Geom. Topol., 10 (2010), 2277–2314.

    Article  MathSciNet  MATH  Google Scholar 

  50. K. Bredies, M. Holler, M. Storath and A. Weinmann, Total generalized variation for manifold-valued data, SIAM J. Imaging Sci., 11 (2018), 1785–1848.

    Article  MathSciNet  MATH  Google Scholar 

  51. L.M. Brègman, Finding the common point of convex sets by the method of successive projection, Dokl. Akad. Nauk SSSR, 162 (1965), 487–490.

    MathSciNet  Google Scholar 

  52. H. Brézis, Opérateurs Maximaux Monotones Et Semi-Groupes De Contractions Dans Les Espaces De Hilbert, North-Holland Math. Stud., 5, North-Holland Publ. Co., Amsterdam, 1973.

    MATH  Google Scholar 

  53. H. Brézis and P.-L. Lions, Produits infinis de résolvantes, Israel J. Math., 29 (1978), 329–345.

    Article  MathSciNet  MATH  Google Scholar 

  54. H. Brézis and A. Pazy, Semigroups of nonlinear contractions on convex sets, J. Functional Analysis, 6 (1970), 237–281.

    Article  MathSciNet  MATH  Google Scholar 

  55. H. Brézis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Functional Analysis, 9 (1972), 63–74.

    Article  MathSciNet  MATH  Google Scholar 

  56. M.R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren Math. Wiss., 319, Springer-Verlag, 1999.

  57. F.E. Browder and W.V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc., 72 (1966), 571–575.

    Article  MathSciNet  MATH  Google Scholar 

  58. D.G. Brown and M. Owen, Mean and variance of phylogenetic trees, Systematic Biology, 69 (2019), 139–154.

    Article  Google Scholar 

  59. R.E. Bruck, Jr., Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Functional Analysis, 18 (1975), 15–26.

    Article  MathSciNet  MATH  Google Scholar 

  60. H. Busemann, Spaces with non-positive curvature, Acta Math., 80 (1948), 259–310.

    Article  MathSciNet  MATH  Google Scholar 

  61. J. Chalopin, V. Chepoi, H. Hirai and D. Osajda, Weakly Modular Graphs and Nonpositive Curvature, Mem. Amer. Math. Soc., 268, no. 1309, Amer. Math. Soc., Providence, RI, 2020.

    MATH  Google Scholar 

  62. J. Chalopin, V. Chepoi and G. Naves, Isometric embedding of Busemann surfaces into Li, Discrete Comput. Geom., 53 (2015), 16–37.

    Article  MathSciNet  MATH  Google Scholar 

  63. V. Chepoi, B. Estellon and G. Naves, Packing and covering with balls on Busemann surfaces, Discrete Comput. Geom., 57 (2017), 985–1011.

    Article  MathSciNet  MATH  Google Scholar 

  64. V. Chepoi and D. Maftuleac, Shortest path problem in rectangular complexes of global nonpositive curvature, Comput. Geom., 46 (2013), 51–64.

    Article  MathSciNet  MATH  Google Scholar 

  65. P.R. Chernoff, Note on product formulas for operator semigroups, J. Functional Analysis, 2 (1968), 238–242.

    Article  MathSciNet  MATH  Google Scholar 

  66. P. Clément and J. Maas, A Trotter product formula for gradient flows in metric spaces, J. Evol. Equ., 11 (2011), 405–427.

    Article  MathSciNet  MATH  Google Scholar 

  67. P.L. Combettes and L.E. Glaudin, Proximal activation of smooth functions in splitting algorithms for convex image recovery, SIAM J. Imaging Sci., 12 (2019), 1905–1935.

    Article  MathSciNet  MATH  Google Scholar 

  68. M.G. Crandall and T.M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265–298.

    Article  MathSciNet  MATH  Google Scholar 

  69. G. Dal Maso, An Introduction to Γ-Convergence, Progr. Nonlinear Differential Equations Appl., 8, Birkhäuser Boston Inc., Boston, MA, 1993.

    Book  MATH  Google Scholar 

  70. A. Daniilidis, G. David, E. Durand-Cartagena and A. Lemenant, Rectifiability of self-contracted curves in the Euclidean space and applications, J. Geom. Anal., 25 (2015), 1211–1239.

    Article  MathSciNet  MATH  Google Scholar 

  71. A. Daniilidis, O. Ley and S. Sabourau, Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions, J. Math. Pures Appl. (9), 94 (2010), 183–199.

    Article  MathSciNet  MATH  Google Scholar 

  72. S.K. Donaldson, Conjectures in Kähler geometry, In: Strings and Geometry, Clay Math. Proc., 3, Amer. Math. Soc., Providence, RI, 2004, pp. 71–78.

    Google Scholar 

  73. D. Drusvyatskiy, The proximal point method revisited, preprint, ar**v:1712.06038.

  74. B. Duchesne, Groups acting on spaces of non-positive curvature, In: Handbook of Group Actions. Vol. III, Adv. Lect. Math. (ALM), 40, Int. Press, Somerville, MA, 2018, pp. 101–141.

    MATH  Google Scholar 

  75. M. Edelstein, The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc., 78 (1972), 206–208.

    Article  MathSciNet  MATH  Google Scholar 

  76. P. Enflo, On the nonexistence of uniform homeomorphisms between Lp-spaces, Ark. Mat., 8 (1969), 103–105.

    Article  MathSciNet  MATH  Google Scholar 

  77. P. Enflo, On infinite-dimensional topological groups, In: Séminaire sur la Géométrie des Espaces de Banach (1977–1978), École Polytech., Palaiseau, 1978, no. 10–11.

    Google Scholar 

  78. A. Eskenazis, M. Mendel and A. Naor, Nonpositive curvature is not coarsely universal, Invent. Math., 217 (2019), 833–886.

    Article  MathSciNet  MATH  Google Scholar 

  79. R. Espínola and A. Fernández-León, CAT(k)-spaces, weak convergence and fixed points, J. Math. Anal. Appl., 353 (2009), 410–427.

    Article  MathSciNet  MATH  Google Scholar 

  80. R. Espínola and A. Nicolae, Proximal minimization in CAT(κ) spaces, J. Nonlinear Convex Anal., 17 (2016), 2329–2338.

    MathSciNet  MATH  Google Scholar 

  81. M. Fabian, P. Habala, P. Hájek, V. Montesinos and V. Zizler, Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books Math./Ouvrages Math. SMC, Springer-Verlag, 2011.

  82. O.P. Ferreira and P.R. Oliveira, Proximal point algorithm on Riemannian manifolds, Optimization, 51 (2002), 257–270.

    Article  MathSciNet  MATH  Google Scholar 

  83. T. Foertsch, A. Lytchak and V. Schroeder, Nonpositive curvature and the Ptolemy inequality, Int. Math. Res. Not. IMRN, 2007 (2007), rnm100.

    MathSciNet  MATH  Google Scholar 

  84. B. Fuglede, Harmonic maps from Riemannian polyhedra to geodesic spaces with curvature bounded from above, Calc. Var. Partial Differential Equations, 31 (2008), 99–136.

    Article  MathSciNet  MATH  Google Scholar 

  85. B. Fuglede, Homotopy problems for harmonic maps to spaces of nonpositive curvature, Comm. Anal. Geom., 16 (2008), 681–733.

    Article  MathSciNet  MATH  Google Scholar 

  86. S. Fujishige, Submodular Functions and Optimization. Second ed., Ann. Discrete Math., 58, Elsevier B.V., Amsterdam, 2005.

    MATH  Google Scholar 

  87. A. Gavryushkin and A.J. Drummond, The space of ultrametric phylogenetic trees, J. Theoret. Biol., 403 (2016), 197–208.

    Article  MathSciNet  MATH  Google Scholar 

  88. T. Gelander, A. Karlsson and G.A. Margulis, Superrigidity, generalized harmonic maps and uniformly convex spaces, Geom. Funct. Anal., 17 (2008), 1524–1550.

    Article  MathSciNet  MATH  Google Scholar 

  89. K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., 28, Cambridge Univ. Press, Cambridge, 1990.

    Book  MATH  Google Scholar 

  90. K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive map**s, Monogr. Textbooks Pure Appl. Math., 83, Marcel Dekker Inc., New York, 1984.

    MATH  Google Scholar 

  91. M. Gromov, Asymptotic invariants of infinite groups, In: Geometric Group Theory. Vol. 2, London Math. Soc. Lecture Note Ser., 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295.

    Google Scholar 

  92. M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progr. Math., 152, Birkhäuser Boston Inc., Boston, MA, 1999. Based on the 1981 French original, with appendices by M. Katz, P. Pansu and S. Semmes, translated from the French by Sean Michael Bates.

    MATH  Google Scholar 

  93. M. Gromov, CAT(κ)-spaces: construction and concentration, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 280 (2001), 101–140.

    Google Scholar 

  94. M. Gromov, Random walk in random groups, Geom. Funct. Anal., 13 (2003), 73–146.

    Article  MathSciNet  MATH  Google Scholar 

  95. M. Gromov and R. Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math., 76 (1992), 165–246.

    Article  MathSciNet  MATH  Google Scholar 

  96. O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), 403–419.

    Article  MathSciNet  MATH  Google Scholar 

  97. M.J. Gursky and J. Streets, A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow, Geom. Flows, 4 (2019), 30–50.

    Article  MathSciNet  MATH  Google Scholar 

  98. T. Haettel, D. Kielak and P. Schwer, The 6-strand braid group is CAT(0), Geom. Dedicata, 182 (2016), 263–286.

    Article  MathSciNet  MATH  Google Scholar 

  99. M. Hamada and H. Hirai, Maximum vanishing subspace problem, CAT(0)-space relaxation, and block-triangularization of partitioned matrix, preprint, ar**v:1705.02060.

  100. M. Hamada and H. Hirai, Computing the nc-rank via discrete convex optimization on CAT(0) spaces, SIAM J. Appl. Algebra Geom., 5 (2021), 455–478.

    Article  MathSciNet  MATH  Google Scholar 

  101. K. Hayashi, A polynomial time algorithm to compute geodesics in CAT(0) cubical complexes, In: 45th International Colloquium on Automata, Languages, and Programming, LIPIcs. Leibniz Int. Proc. Inform., 107, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018, no. 78.

    Google Scholar 

  102. H. Hirai, L-convexity on graph structures, J. Oper. Res. Soc. Japan, 61 (2018), 71–109.

    MathSciNet  MATH  Google Scholar 

  103. H. Hirai, A nonpositive curvature property of modular semilattices, Geom. Dedicata, 214 (2021), 427–463.

    Article  MathSciNet  MATH  Google Scholar 

  104. H.S. Hundal, An alternating projection that does not converge in norm, Nonlinear Anal., 57 (2004), 35–61.

    Article  MathSciNet  MATH  Google Scholar 

  105. P. Ivanisvili, R. van Handel and A. Volberg, Rademacher type and Enflo type coincide, Ann. of Math. (2), 192 (2020), 665–678.

    Article  MathSciNet  MATH  Google Scholar 

  106. S. Ivanov and A. Lytchak, Rigidity of Busemann convex Finsler metrics, Comment. Math. Helv., 94 (2019), 855–868.

    Article  MathSciNet  MATH  Google Scholar 

  107. H. Izeki, T. Kondo and S. Nayatani, N-step energy of maps and the fixed-point property of random groups, Groups Geom. Dyn., 6 (2012), 701–736.

    Article  MathSciNet  MATH  Google Scholar 

  108. H. Izeki and S. Nayatani, Combinatorial harmonic maps and discrete-group actions on Hadamard spaces, Geom. Dedicata, 114 (2005), 147–188.

    Article  MathSciNet  MATH  Google Scholar 

  109. J. Jost, Equilibrium maps between metric spaces, Calc. Var. Partial Differential Equations, 2 (1994), 173–204.

    Article  MathSciNet  MATH  Google Scholar 

  110. J. Jost, Convex functionals and generalized harmonic maps into spaces of non-positive curvature, Comment. Math. Helv., 70 (1995), 659–673.

    Article  MathSciNet  MATH  Google Scholar 

  111. J. Jost, Generalized Dirichlet forms and harmonic maps, Calc. Var. Partial Differential Equations, 5 (1997), 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  112. J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects, Lectures Math. ETH Zürich, Birkhäuser Verlag, Basel, 1997.

    Book  MATH  Google Scholar 

  113. J. Jost, Nonlinear Dirichlet forms, In: New Directions in Dirichlet Forms, AMS/IP Stud. Adv. Math., 8, Amer. Math. Soc., Providence, RI, 1998, pp. 1–47.

    Chapter  MATH  Google Scholar 

  114. S. Kakutani, Some characterizations of Euclidean space, Jpn. J. Math., 16 (1939), 93–97.

    Article  MathSciNet  MATH  Google Scholar 

  115. T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19 (1967), 508–520.

    Article  MathSciNet  MATH  Google Scholar 

  116. T. Kato and K. Masuda, Trotter’s product formula for nonlinear semigroups generated by the subdifferentials of convex functionals, J. Math. Soc. Japan, 30 (1978), 169–178.

    Article  MathSciNet  MATH  Google Scholar 

  117. M. Kell, Symmetric orthogonality and non-expansive projections in metric spaces, Manuscripta Math., 161 (2020), 141–159.

    Article  MathSciNet  MATH  Google Scholar 

  118. Y. Kimura and F. Kohsaka, Spherical nonspreadingness of resolvents of convex functions in geodesic spaces, J. Fixed Point Theory Appl., 18 (2016), 93–115.

    Article  MathSciNet  MATH  Google Scholar 

  119. Y. Kimura and F. Kohsaka, Two modified proximal point algorithms for convex functions in Hadamard spaces, Linear Nonlinear Anal., 2 (2016), 69–86.

    MathSciNet  MATH  Google Scholar 

  120. Y. Kimura and F. Kohsaka, The proximal point algorithm in geodesic spaces with curvature bounded above, Linear Nonlinear Anal., 3 (2017), 133–148.

    MathSciNet  MATH  Google Scholar 

  121. Y. Kimura and F. Kohsaka, Two modified proximal point algorithms in geodesic spaces with curvature bounded above, Rend. Circ. Mat. Palermo (2), 68 (2019), 83–104.

    Article  MathSciNet  MATH  Google Scholar 

  122. W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689–3696.

    Article  MathSciNet  MATH  Google Scholar 

  123. B.R. Kloeckner, Yet another short proof of Bourgain’s distortion estimate for embedding of trees into uniformly convex Banach spaces, Israel J. Math., 200 (2014), 419–422.

    Article  MathSciNet  MATH  Google Scholar 

  124. Y. Kobayashi, Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups, J. Math. Soc. Japan, 27 (1975), 640–665.

    Article  MathSciNet  MATH  Google Scholar 

  125. U. Kohlenbach, Applied Proof Theory: Proof Interpretations and their Use in Mathematics, Springer Monogr. Math., Springer-Verlag, 2008.

  126. U. Kohlenbach, Recent progress in proof mining in nonlinear analysis, IFCoLog J. Log. Appl., 4 (2017), 3357–3406.

    MathSciNet  Google Scholar 

  127. U. Kohlenbach, A polynomial rate of asymptotic regularity for compositions of projections in Hilbert space, Found. Comput. Math., 19 (2019), 83–99.

    Article  MathSciNet  MATH  Google Scholar 

  128. U. Kohlenbach and L. Leuştean, Effective metastability of Halpern iterates in CAT(0) spaces, Adv. Math., 231 (2012), 2526–2556.

    Article  MathSciNet  MATH  Google Scholar 

  129. U. Kohlenbach and L. Leuştean, Addendum to “Effective metastability of Halpern iterates in CAT(0) spaces”, Adv. Math., 250 (2014), 650–651.

    Article  MathSciNet  MATH  Google Scholar 

  130. U. Kohlenbach, L. Leuştean and A. Nicolae, Quantitative results on Fejér monotone sequences, Commun. Contemp. Math., 20 (2018), 1750015.

    Article  MathSciNet  MATH  Google Scholar 

  131. F. Kohsaka, Existence and approximation of fixed points of vicinal map**s in geodesic spaces, Pure Appl. Funct. Anal., 3 (2018), 91–106.

    MathSciNet  MATH  Google Scholar 

  132. T. Kondo, CAT(0) spaces and expanders, Math. Z., 271 (2012), 343–355.

    Article  MathSciNet  MATH  Google Scholar 

  133. E. Kopecká and S. Reich, Nonexpansive retracts in Banach spaces, In: Fixed Point Theory and Its Applications, Banach Center Publ., 77, Polish Acad. Sci. Inst. Math., Warsaw, 2007, pp. 161–174.

    Chapter  Google Scholar 

  134. N.J. Korevaar and R.M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom., 1 (1993), 561–659.

    Article  MathSciNet  MATH  Google Scholar 

  135. N.J. Korevaar and R.M. Schoen, Global existence theorems for harmonic maps to non-locally compact spaces, Comm. Anal. Geom., 5 (1997), 333–387.

    Article  MathSciNet  MATH  Google Scholar 

  136. L.V. Kovalev, Lipschitz retraction of finite subsets of Hilbert spaces, Bull. Aust. Math. Soc., 93 (2016), 146–151.

    Article  MathSciNet  MATH  Google Scholar 

  137. A. Kristály and L. Kozma, Metric characterization of Berwald spaces of nonpositive flag curvature, J. Geom. Phys., 56 (2006), 1257–1270.

    Article  MathSciNet  MATH  Google Scholar 

  138. A. Kristály and A.Róth, Testing metric relations on Finsler manifolds via a geodesic detecting algorithm, In: 2014 IEEE 9th IEEE International Symposium on Applied Computaional Intelligence and Informatics, 2014, pp. 331–336.

  139. A. Kristály, C. Varga and L. Kozma, The dispersing of geodesics in Berwald spaces of non-positive flag curvature, Houston J. Math., 30 (2004), 413–420.

    MathSciNet  MATH  Google Scholar 

  140. H.J. Kushner and G.G. Yin, Stochastic Approximation and Recursive Algorithms and Applications. Second ed., Appl. Math. (N. Y.), 35, Stoch. Model. Appl. Probab., Springer-Verlag, 2003.

  141. K. Kuwae and T. Shioya, Variational convergence over metric spaces, Trans. Amer. Math. Soc., 360 (2008), 35–75.

    Article  MathSciNet  MATH  Google Scholar 

  142. J.-F. Lafont and S. Prassidis, Roundness properties of groups, Geom. Dedicata, 117 (2006), 137–160.

    Article  MathSciNet  MATH  Google Scholar 

  143. U. Lang, B. Pavlović and V. Schroeder, Extensions of Lipschitz maps into Hadamard spaces, Geom. Funct. Anal., 10 (2000), 1527–1553.

    Article  MathSciNet  MATH  Google Scholar 

  144. U. Lang and V. Schroeder, Kirszbraun’s theorem and metric spaces of bounded curvature, Geom. Funct. Anal., 7 (1997), 535–560.

    Article  MathSciNet  MATH  Google Scholar 

  145. N. Lebedeva and A. Petrunin, 5-point CAT(0) spaces after Tetsu Toyoda, Anal. Geom. Metr. Spaces, 9 (2021), 160–166.

    Article  MathSciNet  MATH  Google Scholar 

  146. L. Leuştean, A. Nicolae and A. Sipoş, An abstract proximal point algorithm, J. Global Optim., 72 (2018), 553–577.

    Article  MathSciNet  MATH  Google Scholar 

  147. L. Leuştean and A. Sipoş, An application of proof mining to the proximal point algorithm in CAT(0) spaces, In: Mathematics Almost Everywhere, World Sci. Publ., Hackensack, NJ, 2018, pp. 153–167.

    Chapter  Google Scholar 

  148. C. Li, G. López and V. Martín-Márquez, Monotone vector fields and the proximal point algorithm on Hadamard manifolds, J. Lond. Math. Soc. (2), 79 (2009), 663–683.

    Article  MathSciNet  MATH  Google Scholar 

  149. C. Li, G. López, V. Martín-Márquez and J.-H. Wang, Resolvents of set-valued monotone vector fields in Hadamard manifolds, Set-Valued Var. Anal., 19 (2011), 361–383.

    Article  MathSciNet  MATH  Google Scholar 

  150. B. Lin, B. Sturmfels, X. Tang and R. Yoshida, Convexity in tree spaces, SIAM J. Discrete Math., 31 (2017), 2015–2038.

    Article  MathSciNet  MATH  Google Scholar 

  151. N. Linial and M. Saks, The Euclidean distortion of complete binary trees, Discrete Comput. Geom., 29 (2003), 19–21.

    Article  MathSciNet  MATH  Google Scholar 

  152. A. Lytchak, Open map theorem for metric spaces, Algebra i Analiz, 17 (2005), 139–159.

    MathSciNet  Google Scholar 

  153. A. Lytchak and A. Petrunin, Weak topology on CAT(0) spaces, preprint, ar**v:2107.09295.

  154. A. Lytchak and A. Petrunin, Cyclic projections in Hadamard spaces, J. Optim. Theory Appl., 194 (2022), 636–642.

    Article  MathSciNet  MATH  Google Scholar 

  155. A. Lytchak and S. Stadler, Improvements of upper curvature bounds, Trans. Amer. Math. Soc., 373 (2020), 7153–7166.

    Article  MathSciNet  MATH  Google Scholar 

  156. B. Martinet, Régularisation d’inéquations variationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle, 4 (1970), 154–158.

    MathSciNet  MATH  Google Scholar 

  157. J. Matoušek, On embedding trees into uniformly convex Banach spaces, Israel J. Math., 114 (1999), 221–237.

    Article  MathSciNet  MATH  Google Scholar 

  158. U.F. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom., 6 (1998), 199–253.

    Article  MathSciNet  MATH  Google Scholar 

  159. M. Mendel and A. Naor, Expanders with respect to Hadamard spaces and random graphs, Duke Math. J., 164 (2015), 1471–1548.

    Article  MathSciNet  MATH  Google Scholar 

  160. C. Mese, Uniqueness theorems for harmonic maps into metric spaces, Commun. Contemp. Math., 4 (2002), 725–750.

    Article  MathSciNet  MATH  Google Scholar 

  161. E. Miller, Fruit flies and moduli: interactions between biology and mathematics, Notices Amer. Math. Soc., 62 (2015), 1178–1184.

    Article  MathSciNet  MATH  Google Scholar 

  162. E. Miller, M. Owen and J.S. Provan, Polyhedral computational geometry for averaging metric phylogenetic trees, Adv. in Appl. Math., 68 (2015), 51–91.

    Article  MathSciNet  MATH  Google Scholar 

  163. I. Miyadera and S. Ôharu, Approximation of semi-groups of nonlinear operators, Tohoku Math. J. (2), 22 (1970), 24–47.

    Article  MathSciNet  MATH  Google Scholar 

  164. N. Monod, Superrigidity for irreducible lattices and geometric splitting, J. Amer. Math. Soc., 19 (2006), 781–814.

    Article  MathSciNet  MATH  Google Scholar 

  165. N. Monod, Extreme points in non-positive curvature, Studia Math., 234 (2016), 265–270.

    MathSciNet  MATH  Google Scholar 

  166. J.-J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien, C. R. Acad. Sci. Paris, 255 (1962), 2897–2899.

    MathSciNet  MATH  Google Scholar 

  167. J.-J. Moreau, Propriétés des applications “prox”, C. R. Acad. Sci. Paris, 256 (1963), 1069–1071.

    MathSciNet  MATH  Google Scholar 

  168. J.-J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273–299.

    Article  MathSciNet  MATH  Google Scholar 

  169. U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510–585.

    Article  MathSciNet  MATH  Google Scholar 

  170. M. Movahedi, D. Behmardi and S. Hosseini, On the density theorem for the subdifferential of convex functions on Hadamard spaces, Pacific J. Math., 276 (2015), 437–447.

    Article  MathSciNet  MATH  Google Scholar 

  171. A. Navas, An L1 ergodic theorem with values in a non-positively curved space via a canonical barycenter map, Ergodic Theory Dynam. Systems, 33 (2013), 609–623.

    Article  MathSciNet  MATH  Google Scholar 

  172. S. Neumayer, J. Persch and G. Steidl, Morphing of manifold-valued images inspired by discrete geodesics in image spaces, SIAM J. Imaging Sci., 11 (2018), 1898–1930.

    Article  MathSciNet  MATH  Google Scholar 

  173. T.M.W. Nye, Convergence of random walks to Brownian motion on cubical complexes, preprint, ar**v:1508.02906.

  174. T.M.W. Nye, X. Tang, G. Weyenberg and R. Yoshida, Principal component analysis and the locus of the Fréchet mean in the space of phylogenetic trees, Biometrika, 104 (2017), 901–922.

    Article  MathSciNet  MATH  Google Scholar 

  175. S. Ohta, Harmonicity of totally geodesic maps into nonpositively curved metric spaces, Manuscripta Math., 114 (2004), 127–138.

    Article  MathSciNet  MATH  Google Scholar 

  176. S. Ohta, Markov type of Alexandrov spaces of non-negative curvature, Mathematika, 55 (2009), 177–189.

    Article  MathSciNet  MATH  Google Scholar 

  177. S. Ohta, Self-contracted curves in CAT(0)-spaces and their rectifiability, J. Geom. Anal., 30 (2020), 936–967.

    Article  MathSciNet  MATH  Google Scholar 

  178. S. Ohta and M. Pálfia, Discrete-time gradient flows and law of large numbers in Alexandrov spaces, Calc. Var. Partial Differential Equations, 54 (2015), 1591–1610.

    Article  MathSciNet  MATH  Google Scholar 

  179. S. Ohta and M. Pálfia, Gradient flows and a Trotter–Kato formula of semi-convex functions on CAT(1)-spaces, Amer. J. Math., 139 (2017), 937–965.

    Article  MathSciNet  MATH  Google Scholar 

  180. M. Owen, Computing geodesic distances in tree space, SIAM J. Discrete Math., 25 (2011), 1506–1529.

    Article  MathSciNet  MATH  Google Scholar 

  181. M. Owen and J.S. Provan, A fast algorithm for computing geodesic distances in tree space, IEEE/ACM Trans. Comput. Biol. Bioinform., 8 (2011), 2–13.

    Article  Google Scholar 

  182. E.A. Papa Quiroz and P.R. Oliveira, Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds, J. Convex Anal., 16 (2009), 49–69.

    MathSciNet  MATH  Google Scholar 

  183. N. Parikh and S. Boyd, Proximal algorithms, Found. Trends Optim., 1 (2013), 123–231.

    Google Scholar 

  184. X. Pennec, P. Fillard and N. Ayache, A Riemannian framework for tensor computing, Int. J. Comput. Vis., 66 (2006), 41–66.

    Article  MATH  Google Scholar 

  185. G. Perelman and A. Petrunin, Quasigeodesics and gradient curves in Alexandrov spaces, preprint, 1994.

  186. A. Petrunin, Applications of quasigeodesics and gradient curves, In: Comparison Geometry, Math. Sci. Res. Inst. Publ., 30, Cambridge Univ. Press, Cambridge, 1997, pp. 203–219.

    Google Scholar 

  187. J. Peypouquet and S. Sorin, Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time, J. Convex Anal., 17 (2010), 1113–1163.

    MathSciNet  MATH  Google Scholar 

  188. R.R. Phelps, Convex sets and nearest points, Proc. Amer. Math. Soc., 8 (1957), 790–797.

    Article  MathSciNet  MATH  Google Scholar 

  189. S. Reich, Product formulas, nonlinear semigroups, and accretive operators, J. Functional Analysis, 36 (1980), 147–168.

    Article  MathSciNet  MATH  Google Scholar 

  190. S. Reich, A complement to Trotter’s product formula for nonlinear semigroups generated by the subdifferentials of convex functionals, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), 193–195.

    Article  MathSciNet  MATH  Google Scholar 

  191. S. Reich, Solutions of two problems of H. Brézis, J. Math. Anal. Appl., 95 (1983), 243–250.

    Article  MathSciNet  MATH  Google Scholar 

  192. S. Reich, The asymptotic behavior of a class of nonlinear semigroups in the Hilbert ball, J. Math. Anal. Appl., 157 (1991), 237–242.

    Article  MathSciNet  MATH  Google Scholar 

  193. S. Reich and Z. Salinas, Weak convergence of infinite products of operators in Hadamard spaces, Rend. Circ. Mat. Palermo (2), 65 (2016), 55–71.

    Article  MathSciNet  MATH  Google Scholar 

  194. S. Reich and Z. Salinas, Metric convergence of infinite products of operators in Hadamard spaces, J. Nonlinear Convex Anal., 18 (2017), 331–345.

    MathSciNet  MATH  Google Scholar 

  195. S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal., 15 (1990), 537–558.

    Article  MathSciNet  MATH  Google Scholar 

  196. S. Reich and D. Shoikhet, Semigroups and generators on convex domains with the hyperbolic metric, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 8 (1997), 231–250.

    MathSciNet  MATH  Google Scholar 

  197. J.G. Rešetnjak, Non-expansive maps in a space of curvature no greater than K, Sibirsk. Mat. Ž., 9 (1968), 918–927.

    MathSciNet  Google Scholar 

  198. H. Robbins and D. Siegmund, A convergence theorem for non negative almost supermartingales and some applications, In: Optimizing Methods in Statistics, Proc. Sympos., Ohio State Univ., Columbus, Ohio, 1971, Academic Press, New York, 1971, pp. 233–257.

    Google Scholar 

  199. R.T. Rockafellar, Integrals which are convex functionals, Pacific J. Math., 24 (1968), 525–539.

    Article  MathSciNet  MATH  Google Scholar 

  200. R.T. Rockafellar, Convex integral functionals and duality, In: Contributions to Nonlinear Functional Analysis, Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, 1971, Academic Press, New York, 1971, pp. 215–236.

    Chapter  Google Scholar 

  201. R.T. Rockafellar, Integrals which are convex functionals. II, Pacific J. Math., 39 (1971), 439–469.

    Article  MathSciNet  MATH  Google Scholar 

  202. R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877–898.

    Article  MathSciNet  MATH  Google Scholar 

  203. 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.

    Article  MathSciNet  MATH  Google Scholar 

  204. S. Skwerer, S. Provan and J.S. Marron, Relative optimality conditions and algorithms for treespace Fréchet means, SIAM J. Optim., 28 (2018), 959–988.

    Article  MathSciNet  MATH  Google Scholar 

  205. G. Steidl, Combined first and second order variational approaches for image processing, Jahresber. Dtsch. Math.-Ver., 117 (2015), 133–160.

    Article  MathSciNet  MATH  Google Scholar 

  206. I. Stojkovic, Approximation for convex functionals on non-positively curved spaces and the Trotter–Kato product formula, Adv. Calc. Var., 5 (2012), 77–126.

    Article  MathSciNet  MATH  Google Scholar 

  207. M. Storath and A. Weinmann, Variational regularization of inverse problems for manifold-valued data, Inf. Inference, 10 (2021), 195–230.

    Article  MathSciNet  MATH  Google Scholar 

  208. J. Streets, The consistency and convergence of K-energy minimizing movements, Trans. Amer. Math. Soc., 368 (2016), 5075–5091.

    Article  MathSciNet  MATH  Google Scholar 

  209. K.-T. Sturm, Monotone approximation of energy functionals for map**s into metric spaces. I, J. Reine Angew. Math., 486 (1997), 129–151.

    MathSciNet  MATH  Google Scholar 

  210. K.-T. Sturm, Nonlinear Markov operators associated with symmetric Markov kernels and energy minimizing maps between singular spaces, Calc. Var. Partial Differential Equations, 12 (2001), 317–357.

    Article  MathSciNet  MATH  Google Scholar 

  211. K.-T. Sturm, Nonlinear Markov operators, discrete heat flow, and harmonic maps between singular spaces, Potential Anal., 16 (2002), 305–340.

    Article  MathSciNet  MATH  Google Scholar 

  212. K.-T. Sturm, Probability measures on metric spaces of nonpositive curvature, In: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Paris, 2002, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003, pp. 357–390.

    Google Scholar 

  213. T. Toyoda, An intrinsic characterization of five points in a CAT(0) space, Anal. Geom. Metr. Spaces, 8 (2020), 114–165.

    Article  MathSciNet  MATH  Google Scholar 

  214. H.F. Trotter, Approximation of semi-groups of operators, Pacific J. Math., 8 (1958), 887–919.

    Article  MathSciNet  MATH  Google Scholar 

  215. H.F. Trotter, On the product of semi-groups of operators, Proc. Amer. Math. Soc., 10 (1959), 545–551.

    Article  MathSciNet  MATH  Google Scholar 

  216. A. Wald, Begründung einer koordinatenlosen Differentialgeometrie der Flächen, Erg. Math. Kolloqu., 7 (1936), 24–46.

    MATH  Google Scholar 

  217. M.-T. Wang, Generalized harmonic maps and representations of discrete groups, Comm. Anal. Geom., 8 (2000), 545–563.

    Article  MathSciNet  MATH  Google Scholar 

  218. S. Wenger, Isoperimetric inequalities of Euclidean type in metric spaces, Geom. Funct. Anal., 15 (2005), 534–554.

    Article  MathSciNet  MATH  Google Scholar 

  219. S. Wenger, Plateau’s problem for integral currents in locally non-compact metric spaces, Adv. Calc. Var., 7 (2014), 227–240.

    Article  MathSciNet  MATH  Google Scholar 

  220. A. Willis, Confidence sets for phylogenetic trees, J. Amer. Statist. Assoc., 114 (2019), 235–244.

    Article  MathSciNet  MATH  Google Scholar 

  221. A. Willis and R. Bell, Uncertainty in phylogenetic tree estimates, J. Comput. Graph. Statist., 27 (2018), 542–552.

    Article  MathSciNet  MATH  Google Scholar 

  222. M. **a, On sharp lower bounds for Calabi-type functionals and destabilizing properties of gradient flows, Anal. PDE, 14 (2021), 1951–1976.

    Article  MathSciNet  MATH  Google Scholar 

  223. T. Yokota, Convex functions and barycenter on CAT(1)-spaces of small radii, J. Math. Soc. Japan, 68 (2016), 1297–1323.

    Article  MathSciNet  MATH  Google Scholar 

  224. T. Yokota, Convex functions and p-barycenter on CAT(1)-spaces of small radii, Tsukuba J. Math., 41 (2017), 43–80.

    Article  MathSciNet  MATH  Google Scholar 

  225. T. Yokota, Law of large numbers in CAT(1)-spaces of small radii, Calc. Var. Partial Differential Equations, 57 (2018), no. 35.

  226. A.J. Zaslavski, Inexact proximal point methods in metric spaces, Set-Valued Var. Anal., 19 (2011), 589–608.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

After posting the first version of the present paper to the ar**v on July 3, 2018, I received a large number of useful comments, which helped me to improve the exposition substantially. My thanks go especially to Goulnara Arzhantseva, Victor Chepoi, Bruno Duchesne, Nicola Gigli, Koyo Hayashi, Hiroshi Hirai, Fumiaki Kohsaka, Leonid Kovalev, Alexandru Kristály, Alexander Lytchak, Uwe Mayer, Manor Mendel, Delio Mugnolo, Anton Petrunin, Gabriele Steidl and Ryokichi Tanaka.

I am also very grateful to the referees for suggesting various improvements and pointing out many misprints. Finally, I would like to express my sincere gratitude to numerous colleagues who encouradged and supported me during the review process.

Author information

Authors and Affiliations

  1. Leipzig, Germany

    Miroslav Bačák

Authors
  1. Miroslav Bačák
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miroslav Bačák.

Additional information

Communicated by: Toshiyuki Kobayashi

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bačák, M. Old and new challenges in Hadamard spaces. Jpn. J. Math. 18, 115–168 (2023). https://doi.org/10.1007/s11537-023-1826-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11537-023-1826-0

Keywords and phrases

Mathematics Subject Classification (2020)

Search

Navigation