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JOURNALS // Symmetry, Integrability and Geometry: Methods and Applications // Archive

SIGMA, 2020 Volume 16, 131, 29 pp. (Mi sigma1668)

This article is cited in 1 paper

Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

Annegret Burtschera, Christian Kettererb, Robert J. McCannb, Eric Woolgarc

a Department of Mathematics, IMAPP, Radboud University, PO Box 9010, Postvak 59, 6500 GL Nijmegen, The Netherlands
b Department of Mathematics, University of Toronto, 40 St George St, Toronto Ontario, Canada M5S 2E4
c Department of Mathematical and Statistical Sciences and Theoretical Physics Institute, University of Alberta, Edmonton AB, Canada T6G 2G1

Abstract: Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and – in the Riemannian curvature-dimension (RCD) setting – characterize the cases of equality.

Keywords: curvature-dimension condition, synthetic mean curvature, optimal transport, comparison geometry, diameter bounds, singularity theorems, inscribed radius, inradius bounds, rigidity, measure contraction property.

MSC: 51K10, 53C21, 30L99, 83C75

Received: June 3, 2020; in final form November 21, 2020; Published online December 10, 2020

Language: English

DOI: 10.3842/SIGMA.2020.131



Bibliographic databases:
ArXiv: 2005.07435


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