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arxiv: 2602.05740 · v2 · submitted 2026-02-05 · 🧮 math.DG · math.MG

Busemann and MCP

Tadashi Fujioka , Kenshiro Tashiro This is my paper

Pith reviewed 2026-05-16 06:59 UTC · model grok-4.3

classification 🧮 math.DG math.MG MSC 53C23
keywords Busemann spacesmeasure contraction propertyrigidity theoremsmetric geometrygeodesic completenesssynthetic curvaturenon-collapsing
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The pith

Busemann spaces with MCP measures obey rigidity and structure theorems when geodesically complete or non-collapsed.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies metric spaces satisfying Busemann's curvature condition together with measures that obey the measure contraction property. It proves that geodesic completeness forces these spaces into rigid forms with controlled geometry and that a non-collapse assumption yields global structure results on how the space and measure behave at large scales. A reader would care because the results link a classical metric curvature notion to modern synthetic lower bounds on curvature coming from optimal transport, giving classification tools for spaces that are not too curved from below.

Core claim

Under the assumption of geodesic completeness, Busemann spaces carrying an MCP measure are rigid in the sense that their geometry is determined by the curvature bound; under a non-collapse condition the same data determine the structure of the space, including the form of its tangent cones and the regularity of the measure.

What carries the argument

The Busemann curvature condition on the metric together with the measure contraction property (MCP) on the measure, used to derive rigidity from completeness.

If this is right

  • The tangent cones of geodesically complete examples are themselves Busemann spaces with MCP measures.
  • The measure must be comparable to Hausdorff measure in the non-collapsed case.
  • No branching or splitting can occur beyond what the curvature bound allows.
  • Local Euclidean structure propagates globally under completeness.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same rigidity might hold for the weaker CD condition in place of MCP.
  • The results could classify certain length spaces arising in geometric group theory.
  • Appendix observations on tangent cones suggest a way to study limits without completeness.

Load-bearing premise

The space must be geodesically complete or the measure must satisfy a non-collapse condition for the rigidity and structure conclusions to hold.

What would settle it

An explicit example of a geodesically incomplete Busemann space equipped with an MCP measure that is not rigid or does not satisfy the predicted structure would disprove the main theorems.

Figures

Figures reproduced from arXiv: 2602.05740 by Kenshiro Tashiro, Tadashi Fujioka.

Figure 1
Figure 1. Figure 1: We show m(B(x, r)) ≤ m(B(y, r)). Let Φ be the (1−d(x, y)/d(x, y′ ))-contraction map centered at y ′ (in particular, Φ(x) = y). The measure contraction property (2.3) implies  1 − d(x, y) d(x, y′) N m(B(x, r)) ≤ m(Φ(B(x, r))). (4.1) By the Busemann convexity (2.1), Φ is always 1-Lipschitz, and hence Φ(B(x, r)) ⊂ B(y, r). (4.2) Combining (4.1) and (4.2) and taking d(x, y′ ) → ∞, we obtain m(B(x, r)) ≤ m(B(… view at source ↗
Figure 2
Figure 2. Figure 2: Let D := d(x, y)/2. Next we show m(B(γ(t), D) ∪ B(η(t), D)) ≥ m(B(x, D) ∪ B(y, D)). (4.5) Let λi := 1−t/d(qi , x) and Φi the λi-contraction map centered at qi (in particular, Φi(x) = γ(t) and Φi(y) = hi(t)). Then the measure contraction property (2.3) implies m(Φi(B(x, D) ∪ B(y, D))) ≥ λ N i m(B(x, D) ∪ B(y, D)). (4.6) By the Busemann convexity (2.1) as before, we have Φi(B(x, D) ∪ B(y, D)) ⊂ B(γ(t), D) ∪ … view at source ↗
Figure 3
Figure 3. Figure 3: Proof of Theorem 1.1. Let X be as in Theorem 1.1. By Proposition 4.3, X is of cone-type (at every point) and in particular non-branching (cf. Proposition 3.8). Therefore, Theorem 3.1 implies that X is isometric to a strictly convex Banach space of dimension n for some n. Since the Hausdorff dimension is less than or equal to the dimension parameter N, [63, Corollary 2.7], we have n ≤ N. Furthermore, Lemma … view at source ↗
Figure 4
Figure 4. Figure 4: By the MCP(0, n) condition, we have Hn(Φi(B(p, ri))) s n i ≥ Hn(B(p, ri)) r n i . (5.3) By the definition of si , the image of Φi misses some (δsi/2)-ball that is entirely con￾tained in B(p, si) (indeed, if qi ∈ B(p, si) is not contained in the (δsi)-neighborhood of the image of Φi , then let q ′ i be a point on the shortest path pqi at distance δsi/2 from qi and consider the (δsi/2)-ball around q ′ i ). B… view at source ↗
Figure 5
Figure 5. Figure 5: Suppose t < t0 and let x, y ∈ B(p, tR) (see [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Subdivide γt by finitely many points xt = z0, z1, . . . , zN−1, zN = yt so that any adjacent zi , zi+1 lie in a single Uα (see [PITH_FULL_IMAGE:figures/full_fig_p032_8.png] view at source ↗
read the original abstract

We study the structure of Busemann spaces with measures satisfying the measure contraction property (MCP). The main results are rigidity theorems and structure theorems under the assumption of geodesic completeness or non-collapse. The appendix contains some observations on the tangent cones of geodesically complete Busemann spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies the structure of Busemann spaces equipped with measures satisfying the measure contraction property (MCP). The main results are rigidity theorems and structure theorems under the assumptions of geodesic completeness or non-collapse. The appendix contains observations on the tangent cones of geodesically complete Busemann spaces.

Significance. If the results hold, the work advances the theory of metric measure spaces with synthetic curvature bounds by establishing rigidity and structure theorems in Busemann spaces, which generalize non-positive curvature settings. The explicit hypotheses of geodesic completeness or non-collapse address potential pathologies and strengthen the claims. The appendix on tangent cones provides useful local structure information that supports applications in optimal transport and comparison geometry.

minor comments (3)
  1. [§2] §2: The definition of the Busemann property should include an explicit reference to the original curvature condition to clarify the setting for readers unfamiliar with the metric geometry literature.
  2. [Appendix A] Appendix A: The tangent cone observations would benefit from a short comparison paragraph with known results for Alexandrov spaces or CD(K,N) spaces to better highlight the contribution.
  3. [Introduction] Introduction: A citation to the foundational MCP papers (e.g., Sturm or Lott-Villani) is missing when first introducing the measure contraction property.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; theorems self-contained under stated assumptions

full rationale

The paper states rigidity and structure theorems for Busemann spaces with MCP measures explicitly conditioned on geodesic completeness or non-collapse. These hypotheses are invoked at the outset of each main result to guarantee tangent cones and control measure contraction, following standard metric-measure space techniques. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or smuggled ansatzes appear; the derivations remain independent of the target conclusions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details available from abstract only; cannot list free parameters, axioms, or invented entities.

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Works this paper leans on

76 extracted references · 76 canonical work pages · 2 internal anchors

  1. [1]

    Alexander, V

    S. Alexander, V. Kapovitch, and A. Petrunin.Alexandrov geometry—foundations. Vol. 236. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2024

    work page 2024

  2. [2]

    S. B. Alexander and R. L. Bishop. The Hadamard-Cartan theorem in locally convex metric spaces.Enseign. Math. (2)36.3-4 (1990), 309–320

    work page 1990

  3. [3]

    Ambrosio, N

    L. Ambrosio, N. Gigli, A. Mondino, and T. Rajala. Riemannian Ricci cur- vature lower bounds in metric measure spaces withσ-finite measure.Trans. Amer. Math. Soc.367.7 (2015), 4661–4701

    work page 2015

  4. [4]

    Ambrosio, N

    L. Ambrosio, N. Gigli, and G. Savar´ e. Metric measure spaces with Riemannian Ricci curvature bounded from below.Duke Math. J.163.7 (2014), 1405–1490

    work page 2014

  5. [5]

    P. D. Andreev. Geometric constructions in the class of Busemann nonposi- tively curved spaces.J. Math. Phys. Anal. Geom.5.1 (2009), 25–37, 107

    work page 2009

  6. [6]

    P. D. Andreev. Proof of the Busemann conjecture forG-spaces of nonpositive curvature.Algebra i Analiz26.2 (2014), 1–20. Translation inSt. Petersburg Math. J.26.2 (2015), 193–206

    work page 2014

  7. [7]

    P. D. Andreev. Normed space structure on a BusemannG-space of cone type. Mat. Zametki101.2 (2017), 169–180. Translation inMath. Notes101.1-2 (2017), 193–202

    work page 2017

  8. [8]

    P. D. Andreev and V. V. Starostina. Normed planes in the tangent cone to a chord space of nonpositive curvature.Izv. Vyssh. Uchebn. Zaved. Mat.1 (2019), 3–17. Translation inRussian Math. (Iz. VUZ)63.1 (2019), 1–13

    work page 2019

  9. [9]

    Barilari and L

    D. Barilari and L. Rizzi. Sharp measure contraction property for generalized H-type Carnot groups.Commun. Contemp. Math.20.6 (2018), 1750081, 24

    work page 2018

  10. [10]

    V. N. Berestovski˘ ı. Busemann spaces with upper-bounded Aleksandrov curva- ture.Algebra i Analiz14.5 (2002), 3–18. Translation inSt. Petersburg Math. J.14.5 (2003), 713–723

    work page 2002

  11. [11]

    V. N. Berestovski˘ ı. Busemann’s results, ideas, questions and locally compact homogeneous geodesic spaces.Herbert Busemann Selected works. I, 41–84. Springer, Cham, 2018

    work page 2018

  12. [12]

    V. N. Berestovskij and I. G. Nikolaev. Multidimensional generalized Riemann- ian spaces.Geometry, IV, 165–243, 245–250. Springer, Berlin, 1993. 42 REFERENCES

    work page 1993

  13. [13]

    Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groups, 2024

    S. Borza, M. Magnabosco, T. Rossi, and T. Tashiro. Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groups. Preprint, arXiv:2402.14779v1, 2024

    work page arXiv 2024

  14. [14]

    Borza and K

    S. Borza and K. Tashiro. Measure contraction property, curvature exponent and geodesic dimension of sub-Finslerℓ p-Heisenberg groups. To appear in Ann. Inst. Fourier (Grenoble), arXiv:2305.16722v2, 2025

    work page arXiv 2025

  15. [15]

    B. H. Bowditch. Minkowskian subspaces of non-positively curved metric spaces. Bull. London Math. Soc.27.6 (1995), 575–584

    work page 1995

  16. [16]

    M. R. Bridson and A. Haefliger.Metric spaces of non-positive curvature. Vol. 319. Grundlehren der mathematischen Wissenschaften [Fundamental Prin- ciples of Mathematical Sciences]. Springer-Verlag, Berlin, 1999

    work page 1999

  17. [17]

    Bru´ e and D

    E. Bru´ e and D. Semola. Constancy of the dimension for RCD(K, N) spaces via regularity of Lagrangian flows.Comm. Pure Appl. Math.73.6 (2020), 1141–1204

    work page 2020

  18. [18]

    Burago, Y

    D. Burago, Y. Burago, and S. Ivanov.A course in metric geometry. Vol. 33. Graduate Studies in Mathematics. American Mathematical Society, Provi- dence, RI, 2001

    work page 2001

  19. [19]

    Burago, M

    Y. Burago, M. Gromov, and G. Perel’man. A. D. Aleksandrov spaces with curvatures bounded below.Uspekhi Mat. Nauk47.2(284) (1992), 3–51, 222. Translation inRussian Math. Surveys47.2 (1992), 1–58

    work page 1992

  20. [20]

    Busemann

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

    work page 1948

  21. [21]

    Busemann.The geometry of geodesics

    H. Busemann.The geometry of geodesics. Academic Press, Inc., New York, 1955

    work page 1955

  22. [22]

    Cavalletti and E

    F. Cavalletti and E. Milman. The globalization theorem for the curvature- dimension condition.Invent. Math.226.1 (2021), 1–137

    work page 2021

  23. [23]

    X. Dai, S. Honda, J. Pan, and G. Wei. Singular Weyl’s law with Ricci cur- vature bounded below.Trans. Amer. Math. Soc. Ser. B10(2023), 1212– 1253

    work page 2023

  24. [24]

    Q. Deng. H¨ older continuity of tangent cones in RCD(K, N) spaces and appli- cations to nonbranching.Geom. Topol.29.2 (2025), 1037–1114

    work page 2025

  25. [25]

    Descombes

    D. Descombes. Asymptotic rank of spaces with bicombings.Math. Z.284.3-4 (2016), 947–960

    work page 2016

  26. [26]

    Descombes and U

    D. Descombes and U. Lang. Convex geodesic bicombings and hyperbolicity. Geom. Dedicata177(2015), 367–384

    work page 2015

  27. [27]

    Descombes and U

    D. Descombes and U. Lang. Flats in spaces with convex geodesic bicombings. Anal. Geom. Metr. Spaces4.1 (2016), 68–84

    work page 2016

  28. [28]

    Di Marino, N

    S. Di Marino, N. Gigli, E. Pasqualetto, and E. Soultanis. Infinitesimal Hilber- tianity of locally CAT(κ)-spaces.J. Geom. Anal.31.8 (2021), 7621–7685

    work page 2021

  29. [29]

    Topological regularity of Busemann spaces of non- positive curvature

    T. Fujioka and S. Gu. Topological regularity of Busemann spaces of nonpos- itive curvature. Preprint, arXiv:2504.14455v2, 2025

    work page arXiv 2025

  30. [30]

    T. Fukaya. A topological product decomposition of Busemann space.J. Math. Soc. Japan76.1 (2024), 269–281

    work page 2024

  31. [31]

    Geoghegan and P

    R. Geoghegan and P. Ontaneda. Boundaries of cocompact proper CAT(0) spaces.Topology46.2 (2007), 129–137

    work page 2007

  32. [32]

    N. Gigli. The splitting theorem in non-smooth context. Preprint, arXiv:1302. 5555v1, 2013. REFERENCES 43

    work page 2013

  33. [33]

    N. Gigli. On the differential structure of metric measure spaces and applica- tions.Mem. Amer. Math. Soc.236.1113 (2015), vi+91

    work page 2015

  34. [34]

    M. Gromov. Hyperbolic manifolds, groups and actions.Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), 183–213. Princeton Univ. Press, Princeton, NJ, 1981

    work page 1978

  35. [35]

    Hakavuori and E

    E. Hakavuori and E. Le Donne. Blowups and blowdowns of geodesics in Carnot groups.J. Differential Geom.123.2 (2023), 267–310

    work page 2023

  36. [36]

    On the Structure of Busemann Spaces with Non-Negative Curvature

    B.-X. Han and L. Yin. On the structure of Busemann spaces with non- negative curvature. Preprint, arXiv:2508.12348v2, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  37. [37]

    Hitzelberger and A

    P. Hitzelberger and A. Lytchak. Spaces with many affine functions.Proc. Amer. Math. Soc.135.7 (2007), 2263–2271

    work page 2007

  38. [38]

    S. Honda. Metric measure spaces with Ricci bounds from below.Sugaku Ex- positions37.2 (2024), 179–201. Translation ofS¯ ugaku72.2 (2020), 158–181

    work page 2024

  39. [39]

    Ivanov and A

    S. Ivanov and A. Lytchak. Rigidity of Busemann convex Finsler metrics. Comment. Math. Helv.94.4 (2019), 855–868

    work page 2019

  40. [40]

    F. John. Extremum problems with inequalities as subsidiary conditions.Stud- ies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, 187–204. Interscience Publishers, New York, 1948

    work page 1948

  41. [41]

    N. Juillet. Geometric inequalities and generalized Ricci bounds in the Heisen- berg group.Int. Math. Res. Not. IMRN13 (2009), 2347–2373

    work page 2009

  42. [42]

    Kapovitch, M

    V. Kapovitch, M. Kell, and C. Ketterer. On the structure of RCD spaces with upper curvature bounds.Math. Z.301.4 (2022), 3469–3502

    work page 2022

  43. [43]

    Kapovitch and C

    V. Kapovitch and C. Ketterer. Weakly noncollapsed RCD spaces with upper curvature bounds.Anal. Geom. Metr. Spaces7.1 (2019), 197–211

    work page 2019

  44. [44]

    Kapovitch and C

    V. Kapovitch and C. Ketterer. CD meets CAT.J. Reine Angew. Math.766 (2020), 1–44

    work page 2020

  45. [45]

    M. Kell. Sectional curvature-type conditions on metric spaces.J. Geom. Anal. 29.1 (2019), 616–655

    work page 2019

  46. [46]

    Kent and R

    C. Kent and R. Ricks. Asymptotic cones and boundaries of CAT(0) spaces. Indiana Univ. Math. J.70.4 (2021), 1441–1469

    work page 2021

  47. [47]

    Ketterer and T

    C. Ketterer and T. Rajala. Failure of topological rigidity results for the mea- sure contraction property.Potential Anal.42.3 (2015), 645–655

    work page 2015

  48. [48]

    B. Kleiner. The local structure of length spaces with curvature bounded above.Math. Z.231.3 (1999), 409–456

    work page 1999

  49. [49]

    L. Kramer. On the local structure and the homology of CAT(κ) spaces and Euclidean buildings.Adv. Geom.11.2 (2011), 347–369

    work page 2011

  50. [50]

    Krist´ aly and L

    A. Krist´ aly and L. Kozma. Metric characterization of Berwald spaces of non- positive flag curvature.J. Geom. Phys.56.8 (2006), 1257–1270

    work page 2006

  51. [51]

    Krist´ aly, C

    A. Krist´ aly, C. Varga, and L. Kozma. The dispersing of geodesics in Berwald spaces of non-positive flag curvature.Houston J. Math.30.2 (2004), 413–420

    work page 2004

  52. [52]

    Le Donne

    E. Le Donne. Metric spaces with unique tangents.Ann. Acad. Sci. Fenn. Math.36.2 (2011), 683–694

    work page 2011

  53. [53]

    Le Donne

    E. Le Donne. A metric characterization of Carnot groups.Proc. Amer. Math. Soc.143.2 (2015), 845–849

    work page 2015

  54. [54]

    Lott and C

    J. Lott and C. Villani. Ricci curvature for metric-measure spaces via optimal transport.Ann. of Math. (2)169.3 (2009), 903–991. 44 REFERENCES

    work page 2009

  55. [55]

    Luukkainen and E

    J. Luukkainen and E. Saksman. Every complete doubling metric space carries a doubling measure.Proc. Amer. Math. Soc.126.2 (1998), 531–534

    work page 1998

  56. [56]

    Lytchak and K

    A. Lytchak and K. Nagano. Geodesically complete spaces with an upper cur- vature bound.Geom. Funct. Anal.29.1 (2019), 295–342

    work page 2019

  57. [57]

    Lytchak and K

    A. Lytchak and K. Nagano. Topological regularity of spaces with an upper curvature bound.J. Eur. Math. Soc. (JEMS)24.1 (2022), 137–165

    work page 2022

  58. [58]

    Lytchak and V

    A. Lytchak and V. Schroeder. Affine functions on CAT(κ)-spaces.Math. Z. 255.2 (2007), 231–244

    work page 2007

  59. [59]

    Magnabosco

    M. Magnabosco. Examples ofCD(0, N) spaces with non-constant dimension. To appear inAnn. Sc. Norm. Super. Pisa Cl. Sci., arXiv:2310 . 05738v1, 2023

    work page 2023

  60. [60]

    On the rectifiability of CD(K, N) and MCP(K, N) spaces with unique tangents

    M. Magnabosco, A. Mondino, and T. Rossi. On the rectifiability ofCD(K, N) andMCP(K, N) with unique tangents. Preprint, arXiv:2505.01151v1, 2025

    work page arXiv 2025

  61. [61]

    Mondino and A

    A. Mondino and A. Naber. Structure theory of metric measure spaces with lower Ricci curvature bounds.J. Eur. Math. Soc. (JEMS)21.6 (2019), 1809– 1854

    work page 2019

  62. [62]

    Universal non-CD of sub-Riemannian manifolds

    D. Navarro and J. Pan. Universal non-CD of sub-Riemannian manifolds. To appear inJ. Reine Angew. Math., arXiv:2507.00471v2, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  63. [63]

    S.-i. Ohta. On the measure contraction property of metric measure spaces. Comment. Math. Helv.82.4 (2007), 805–828

    work page 2007

  64. [64]

    S.-i. Ohta. Finsler interpolation inequalities.Calc. Var. Partial Differential Equations36.2 (2009), 211–249

    work page 2009

  65. [65]

    Ontaneda

    P. Ontaneda. Cocompact CAT(0) spaces are almost geodesically complete. Topology44.1 (2005), 47–62

    work page 2005

  66. [66]

    Papadopoulos.Metric spaces, convexity and non-positive curvature

    A. Papadopoulos.Metric spaces, convexity and non-positive curvature. Sec- ond. Vol. 6. IRMA Lectures in Mathematics and Theoretical Physics. Euro- pean Mathematical Society (EMS), Z¨ urich, 2014

    work page 2014

  67. [67]

    A. V. Pogorelov. RegularG-spaces of H. Busemann.Dokl. Akad. Nauk SSSR 314.1 (1990), 114–118. Translation inSoviet Math. Dokl.42.2 (1991), 356– 359

    work page 1990

  68. [68]

    A. V. Pogorelov. Busemann regularG-spaces.Rev. Math. Math. Phys.10.4 (1998), 1–99

    work page 1998

  69. [69]

    L. Rifford. Ricci curvatures in Carnot groups.Mathematical Control and Re- lated Fields3.4 (2013), 467–487

    work page 2013

  70. [70]

    L. Rizzi. Measure contraction properties of Carnot groups.Calc. Var. Partial Differential Equations55.3 (2016), Art. 60, 20

    work page 2016

  71. [71]

    L. Rizzi. A counterexample to gluing theorems for MCP metric measure spaces.Bull. Lond. Math. Soc.50.5 (2018), 781–790

    work page 2018

  72. [72]

    Simon.Lectures on Geometric Measure Theory

    L. Simon.Lectures on Geometric Measure Theory. Vol. 3. Proceedings of the Centre for Mathematical Analysis, Australian National University. Australian National University, Centre for Mathematical Analysis, Canberra, 1983

    work page 1983

  73. [73]

    K.-T. Sturm. On the geometry of metric measure spaces. II.Acta Math.196.1 (2006), 133–177

    work page 2006

  74. [74]

    Villani.Optimal transport Old and new

    C. Villani.Optimal transport Old and new. Vol. 338. Grundlehren der math- ematischen Wissenschaften [Fundamental Principles of Mathematical Sci- ences]. Springer-Verlag, Berlin, 2009. REFERENCES 45

    work page 2009

  75. [75]

    A. L. Vol’berg and S. V. Konyagin. On measures with the doubling condition. Izv. Akad. Nauk SSSR Ser. Mat.51.3 (1987), 666–675. Translation inMath. USSR-Izv.30.3 (1988), 629–638

    work page 1987

  76. [76]

    von Renesse

    M.-K. von Renesse. On local Poincar´ e via transportation.Math. Z.259.1 (2008), 21–31. (T. Fujioka)Fukuoka University, Fukuoka 814-0180, Japan Email address:tfujioka@fukuoka-u.ac.jp (K. Tashiro)The University of Osaka, Osaka 560-0043, Japan Email address:tashiro@math.sci.osaka-u.ac.jp

    work page 2008