On the measure contraction property of metric measure spaces

By Shin-ichi Ohta. 2007

252 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/8cb55094d7bfa4243a7b60f3d2823e8a7f540281

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In Applied Mathematics

In Computational Theory and Mathematics

In Radiology, Nuclear Medicine and Imaging

Citations in the corpus

Ricci curvature of graphs (2011, background): “There are some recent works on giving a good notion for a metric measure space to have a “lower Ricci curvature bound”, see [21], [18] and [19].”
Non-collapsed spaces with Ricci curvature bounded from below (2017, unknown): “…by where ∞ 0 t N/ 2 e − t d t coincides for integer N with the volume of the unit ball in R N and Then the Bishop-Gromov inequality, which is valid in the class of MCP ( K, N ) spaces (see [46] and [51] and recall that an RCD ( K, N ) space is also MCP ( K, N )), states that for any x ∈ supp( m ).”
On the topology and the boundary of N–dimensional RCD(K,N) spaces (2019, methodology): “µ1 ∈ P2(X), with µ0,µ1 absolutely continuous with respect to m, any element of OptGeo(µ0,µ1) is concentrated on a set of non-branching geodesics. The deﬁnition of MCP(K,N) given independently by Ohta [Oht07] and Sturm [Stu06b]. On general metric measure spaces the two deﬁnitions slightly diﬀer, but ”
Measure contraction properties of Carnot groups (2015, methodology): “It has been introduced independently by Ohta [16] and Sturm [21].”
Li-Yau inequality on graphs (2013, unknown): “[8, 16, 17, 20, 21, 23]).”
Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator (2011, background): “By now, there exist many notions of generalized curvature, and several of them have found important applications, see Sturm [24], Lott–Villani [19], Ollivier [21], Ohta [20], Bonciocat–Sturm [3], Joulin–Ollivier [14] and the references therein.”
Localization and Tensorization Properties of the Curvature-Dimension Condition for Metric Measure Spaces (2010, background): “ constants) and thus compactness of Mˆ. For the purpose of comparison we point out that a similar, but slightly weaker condition than CD(K,N) – the measure contraction property MCP(K,N) introduced in [Oht07a] and [Stu06b] – satisﬁes the tensorization property due to [Oht07b] (where no assumption of ”
Weak curvature conditions and functional inequalities (2005, methodology): “The measure contraction property is also considered in [ 11 ] and [14]; compare with the proof of Lemma 3.4.”
New formulas for the Laplacian of distance functions and applications (2018, methodology): “l the deﬁnition of the R´enyi Entropy functional EN : P(X) → [0,∞], EN(µ) := Z X ρ1−1/N(x)m(dx), (2.4) where µ= ρm+µs with µs ⊥ m. Next we recall the deﬁnition of MCP(K,N) given independently by Ohta [47] and Sturm [57]. On general metric measure spaces the two deﬁnitions slightly diﬀer, but on esse”
Ricci Curvature on Alexandrov spaces and Rigidity Theorems (2009, background): “If dim(M) = n, Ohta in [20] and Sturm in [38] proved, independently, that M satisfies MCP(n, k) is equivalent to Ric(M) k.”
Cones over metric measure spaces and the maximal diameter theorem (2013, methodology): “7 (Measure contraction property, [36]).”
On local Poincaré via transportation (2005, methodology): “mov-Hausdorﬀ limits of CD(N,k)-spaces will satisfy the (N,k)-measure contraction property. (The corresponding stability result of the (N,K)-MCP condition alone is a little stronger and is obtained in [Oht05] based on [LV04]). 3.2. Segment inequality for (X,d,m). For the precise formulation of the su”
On the geometry of metric measure spaces. II (2006, methodology): “After this paper was completed, we learned of related work by Ohta [ Oh05 ].4”
Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds (2009, unknown): “D G ] 3 1 M ar”