scholarly journals Local curvature-dimension condition implies measure-contraction property

2012 ◽  
Vol 262 (12) ◽  
pp. 5110-5127 ◽  
Author(s):  
Fabio Cavalletti ◽  
Karl-Theodor Sturm
Keyword(s):  
Local Curvature ◽  
Contraction Property ◽  
Measure Contraction Property

scholarly journals Distortion Coefficients of the $$\alpha $$-Grushin Plane

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Samuël Borza
Keyword(s):  
Trigonometric Functions ◽  
Contraction Property ◽  
Grushin Plane ◽  
Measure Contraction Property ◽  
Property Condition

AbstractWe compute the distortion coefficients of the $$\alpha $$ α -Grushin plane. They are expressed in terms of generalised trigonometric functions. Estimates for the distortion coefficients are then obtained and a conjecture of a measure contraction property condition for the generalised Grushin planes is suggested.

scholarly journals Sharp measure contraction property for generalized H-type Carnot groups

2018 ◽  
Vol 20 (06) ◽  
pp. 1750081 ◽  
Author(s):  
Davide Barilari ◽  
Luca Rizzi
Keyword(s):  
Absolute Continuity ◽  
Carnot Group ◽  
Optimal Synthesis ◽  
Carnot Groups ◽  
Contraction Property ◽  
Dimension Formula ◽  
Quadratic Cost ◽  
Measure Contraction Property ◽  
The Absolute

We prove that H-type Carnot groups of rank [Formula: see text] and dimension [Formula: see text] satisfy the [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. The latter integer coincides with the geodesic dimension of the Carnot group. The same result holds true for the larger class of generalized H-type Carnot groups introduced in this paper, and for which we compute explicitly the optimal synthesis. This constitutes the largest class of Carnot groups for which the curvature exponent coincides with the geodesic dimension. We stress that generalized H-type Carnot groups have step 2, include all corank 1 groups and, in general, admit abnormal minimizing curves. As a corollary, we prove the absolute continuity of the Wasserstein geodesics for the quadratic cost on all generalized H-type Carnot groups.

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

2020 ◽  
Author(s):  
Annegret Burtscher ◽  
◽  
Christian Ketterer ◽  
Robert J. McCann ◽  
Eric Woolgar ◽  
...  
Keyword(s):  
Lower Bound ◽  
Mean Curvature ◽  
Measure Space ◽  
Riemannian Curvature ◽  
Metric Measure Spaces ◽  
Contraction Property ◽  
Measure Spaces ◽  
Generalized Mean Curvature ◽  
Measure Contraction Property ◽  
Mean Convex

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.

scholarly journals Optimal maps in essentially non-branching spaces

2017 ◽  
Vol 19 (06) ◽  
pp. 1750007 ◽  
Author(s):  
Fabio Cavalletti ◽  
Andrea Mondino
Keyword(s):  
Measure Space ◽  
Metric Measure Space ◽  
Absolutely Continuous ◽  
Contraction Property ◽  
Transport Plan ◽  
Measure Contraction Property

In this paper we prove that in a metric measure space [Formula: see text] verifying the measure contraction property with parameters [Formula: see text] and [Formula: see text], any optimal transference plan between two marginal measures is induced by an optimal map, provided the first marginal is absolutely continuous with respect to [Formula: see text] and the space itself is essentially non-branching. In particular this shows that there exists a unique transport plan and it is induced by a map.

Sign in / Sign up

Export Citation Format

Share Document