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

Annegret Burtscher ; Christian Ketterer ; Robert J. McCann ; Eric Woolgar

Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

Résumé

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.

Détail

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

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