## A Bishop-type inequality on metric measure spaces with Ricci curvature bounded below

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## Abstract:

We define a Bishop-type inequality on metric measure spaces with Riemannian curvature-dimension condition. The main result in this short article is that any $RCD$ spaces with the Bishop-type inequalities possess only one regular set in not only the measure theoretical sense but also the set theoretical one. As a corollary, the Hausdorff dimension of such $RCD^*(K,N)$ spaces is exactly $N$. We also prove that every tangent cone at any point on such $RCD$ spaces is a metric cone.## References

- Luigi Ambrosio, Nicola Gigli, Andrea Mondino, and Tapio Rajala,
*Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure*, Trans. Amer. Math. Soc.**367**(2015), no. 7, 4661–4701. MR**3335397**, DOI 10.1090/S0002-9947-2015-06111-X - Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré,
*Gradient flows in metric spaces and in the space of probability measures*, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. MR**2401600** - Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré,
*Metric measure spaces with Riemannian Ricci curvature bounded from below*, Duke Math. J.**163**(2014), no. 7, 1405–1490. MR**3205729**, DOI 10.1215/00127094-2681605 - Luigi Ambrosio, Andrea Mondino, and Giuseppe Savaré,
*Nonlinear diffusion equations and curvature conditions in metric measure spaces*, arXiv1509.07273. - Luigi Ambrosio and Paolo Tilli,
*Topics on analysis in metric spaces*, Oxford Lecture Series in Mathematics and its Applications, vol. 25, Oxford University Press, Oxford, 2004. MR**2039660** - Kathrin Bacher and Karl-Theodor Sturm,
*Localization and tensorization properties of the curvature-dimension condition for metric measure spaces*, J. Funct. Anal.**259**(2010), no. 1, 28–56. MR**2610378**, DOI 10.1016/j.jfa.2010.03.024 - Fabio Cavalletti and Karl-Theodor Sturm,
*Local curvature-dimension condition implies measure-contraction property*, J. Funct. Anal.**262**(2012), no. 12, 5110–5127. MR**2916062**, DOI 10.1016/j.jfa.2012.02.015 - Jeff Cheeger and Tobias H. Colding,
*On the structure of spaces with Ricci curvature bounded below. I*, J. Differential Geom.**46**(1997), no. 3, 406–480. MR**1484888** - Jeff Cheeger and Tobias H. Colding,
*On the structure of spaces with Ricci curvature bounded below. II*, J. Differential Geom.**54**(2000), no. 1, 13–35. MR**1815410** - Jeff Cheeger and Tobias H. Colding,
*On the structure of spaces with Ricci curvature bounded below. III*, J. Differential Geom.**54**(2000), no. 1, 37–74. MR**1815411** - Tobias H. Colding,
*Ricci curvature and volume convergence*, Ann. of Math. (2)**145**(1997), no. 3, 477–501. MR**1454700**, DOI 10.2307/2951841 - Tobias Holck Colding and Aaron Naber,
*Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications*, Ann. of Math. (2)**176**(2012), no. 2, 1173–1229. MR**2950772**, DOI 10.4007/annals.2012.176.2.10 - Guido de Philippis, Andrea Marchese, and Filip Rindler,
*On a conjecture of Cheeger*, arXiv:1607.02554v2. - Matthias Erbar, Kazumasa Kuwada, and Karl-Theodor Sturm,
*On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces*, Invent. Math.**201**(2015), no. 3, 993–1071. MR**3385639**, DOI 10.1007/s00222-014-0563-7 - Nicola Gigli,
*The splitting theorem in non-smooth context*, arXiv:1302.5555. - Nicola Gigli and Guido de Philippis,
*From volume cone to metric cone in the nonsmooth setting*, arXiv:1512.03113. - Nicola Gigli, Andrea Mondino, and Tapio Rajala,
*Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below*, J. Reine Angew. Math.**705**(2015), 233–244. MR**3377394**, DOI 10.1515/crelle-2013-0052 - Nicola Gigli, Andrea Mondino, and Giuseppe Savaré,
*Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows*, Proc. Lond. Math. Soc. (3)**111**(2015), no. 5, 1071–1129. MR**3477230**, DOI 10.1112/plms/pdv047 - Nicola Gigli and Enrico Pasqualetto,
*Behaviour of the reference measure on $\mathsf {RCD}$ spaces under charts*, arXiv:1607.05188v1. - Martin Kell,
*A note on Lipschitz continuity of solutions of Poisson equations in metric measure spaces*, arXiv:1307.2224v2. - Martin Kell and Andrea Mondino,
*On the volume measure of non-smooth spaces with Ricci curvature bounded below*, arXiv:1607.02036. - Yu Kitabeppu,
*Lower bound of coarse Ricci curvature on metric measure spaces and eigenvalues of Laplacian*, Geom. Dedicata**169**(2014), 99–107. MR**3175238**, DOI 10.1007/s10711-013-9844-3 - Yu Kitabeppu and Sajjad Lakzian,
*Characterization of low dimensional $RCD^*(K,N)$ spaces*, Anal. Geom. Metr. Spaces**4**(2016), no. 1, 187–215. MR**3550295**, DOI 10.1515/agms-2016-0007 - John Lott and Cédric Villani,
*Ricci curvature for metric-measure spaces via optimal transport*, Ann. of Math. (2)**169**(2009), no. 3, 903–991. MR**2480619**, DOI 10.4007/annals.2009.169.903 - Andrea Mondino and Aaron Naber,
*Structure theory of metric-measure spaces with lower Ricci curvature bounds I*, arXiv:1405.2222v2. - Shin-ichi Ohta,
*Finsler interpolation inequalities*, Calc. Var. Partial Differential Equations**36**(2009), no. 2, 211–249. MR**2546027**, DOI 10.1007/s00526-009-0227-4 - Shin-Ichi Ohta and Karl-Theodor Sturm,
*Heat flow on Finsler manifolds*, Comm. Pure Appl. Math.**62**(2009), no. 10, 1386–1433. MR**2547978**, DOI 10.1002/cpa.20273 - Karl-Theodor Sturm,
*On the geometry of metric measure spaces. I*, Acta Math.**196**(2006), no. 1, 65–131. MR**2237206**, DOI 10.1007/s11511-006-0002-8 - Karl-Theodor Sturm,
*On the geometry of metric measure spaces. II*, Acta Math.**196**(2006), no. 1, 133–177. MR**2237207**, DOI 10.1007/s11511-006-0003-7

## Additional Information

**Yu Kitabeppu**- Affiliation: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
- MR Author ID: 1055218
- Email: y.kitabeppu@gmail.com
- Received by editor(s): April 6, 2016
- Received by editor(s) in revised form: June 28, 2016, and August 14, 2016
- Published electronically: January 6, 2017
- Communicated by: Jeremy Tyson
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 3137-3151 - MSC (2010): Primary 51F99; Secondary 53C20
- DOI: https://doi.org/10.1090/proc/13517
- MathSciNet review: 3637960