(K, N)-Convexity and the Curvature-Dimension Condition for Negative N
By Shin-ichi Ohta. 2013
89 citations recorded by the source corpus.
We extend the range of N to negative values in the (K, N)-convexity (in the sense of Erbar–Kuwada–Sturm), the weighted Ricci curvature $$\mathop {\mathrm {Ric}}\nolimits _N$$RicN and the curvature-dimension condition $$\mathop {\mathrm {CD}}\nolimits (K,N)$$CD(K,N). We generalize a number of results in the case of $$N>0$$N>0 to this setting, including Bochner’s inequality, the Brunn–Minkowski inequality and the equivalence between $$\mathop {\mathrm {Ric}}\nolimits _N \ge K$$RicN≥K and $$\mathop {\mathrm {CD}}\nolimits (K,N)$$CD(K,N). We also show an expansion bound for gradient flows of Lipschitz (K, N)-convex functions.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/9d7bf5ee5bff3e5caf0035b2a393bec2b6d7f24b
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In Applied Mathematics
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- A. V. Kolesnikov
- Zoltán M. Balogh
- Alexandru Krist'aly
- Kolesnikov Alexander
- Milman Emanuel
- Joaquim Martín
- Mario Milman
In Geometry and Topology
- A. V. Kolesnikov
- Emanuel Milman
- Joaquim Martín
- Mario Milman
- Yuansi Chen
- Fabio Cavalletti
- Andrea Mondino
- Dario Cordero-Erausquin
Citations in the corpus
- Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension (2014, methodology): “…was shown by Erbar–Kuwada–Sturm [20] to be equivalent when N > 0 to the usual CD( ρ, N ) condition for essentially non-branching spaces (such as weighted-manifolds, Finsler and Alexandrov spaces), Ohta obtained in [54] variants for N < 0 and ρ > 0 of the HWI, Talagrand and log-Sobolev inequalities.”
- Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary (2013, methodology): “We expect this gap in the literature to be quickly filled (in fact, concurrently to posting our work on the arXiv, Ohta [39] has posted a first attempt of a systematic treatise of the range N ≤ 0, and subsequently other authors have also begun treating this extended range [14,17,31,32,50]).”
- Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds (2017, methodology): “The authors thank Franck Barthe, Bo Berndtsson, Andrea Colesanti, Dario Cordero-Erausquin, Bo’az Klartag, Michel Ledoux, Frank Morgan, Van Hoang Nguyen, Shin-ichi Ohta, Yehuda Pinchover and Steve Zelditch for their comments and interest.”