31 citations · 110 across the 36 of their papers we have counts for
42 papers
A Brenier-Strassen Theorem on CAT(kappa) Spaces
Nathael Gozlan, Hugo Malamut, Shin-Ichi Ohta
We extend the Brenier-Strassen theorem about projections in convex order to non-flat spaces with curvature bounded from above. Precisely, for probability measures , of finit…
Brownian motion in Minkowski normed spaces
Shin-ichi Ohta, Marco Rehmeier, Kohei Suzuki
A Minkowski normed space is the Euclidean space equipped with a (possibly asymmetric) uniformly convex and smooth norm, forming a particular class of Finsler manifolds. We construc…
Discrete-time gradient flows for unbounded convex functions on Gromov hyperbolic spaces
Shin-ichi Ohta
In proper, geodesic Gromov hyperbolic spaces, we investigate discrete-time gradient flows via the proximal point algorithm for unbounded Lipschitz convex functions. Assuming that t…
A generalization of Grünbaum's inequality in RCD-spaces
Victor-Emmanuel Brunel, Shin-ichi Ohta, Jordan Serres
We generalize Grünbaum's classical inequality in convex geometry to curved spaces with nonnegative Ricci curvature, precisely, to -spaces with …
Integral Varadhan formula for non-linear heat flow
Shin-ichi Ohta, Kohei Suzuki
We prove the integral Varadhan short-time formula for non-linear heat flow on measured Finsler manifolds. To the best of the authors' knowledge, this is the first result establishi…
How does the contraction property fail for convex functions on normed spaces?
Shin-ichi Ohta
On Euclidean and Hilbert spaces, Riemannian manifolds, and CAT-spaces, gradient flows of convex functions are known to satisfy the contraction property, which plays a fundamen…
Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes
Mathias Braun, Shin-ichi Ohta
We prove that a Finsler spacetime endowed with a smooth reference measure whose induced weighted Ricci curvature is bounded from below by a real number …
Barycenters and a law of large numbers in Gromov hyperbolic spaces
Shin-ichi Ohta
We investigate barycenters of probability measures on Gromov hyperbolic spaces, toward development of convex optimization in this class of metric spaces. We establish a contraction…
Gradient flows in asymmetric metric spaces and applications
Shin-ichi Ohta, Wei Zhao
This paper is devoted to the investigation of gradient flows in asymmetric metric spaces (for example, irreversible Finsler manifolds and Minkowski normed spaces) by means of discr…
Discrete-time gradient flows in Gromov hyperbolic spaces
Shin-ichi Ohta
We investigate fundamental properties of the proximal point algorithm for Lipschitz convex functions on (proper, geodesic) Gromov hyperbolic spaces. We show that the proximal point…
Quantitative estimates for the Bakry-Ledoux isoperimetric inequality. II
Cong Hung Mai, Shin-ichi Ohta
Concerning quantitative isoperimetry for a weighted Riemannian manifold satisfying , we give an -estimate exhibiting that the push-forward of the…
Geometry of weighted Lorentz-Finsler manifolds II: A splitting theorem
Yufeng Lu, Ettore Minguzzi, Shin-ichi Ohta
We show an analogue of the Lorentzian splitting theorem for weighted Lorentz-Finsler manifolds: If a weighted Berwald spacetime of nonnegative weighted Ricci curvature satisfies ce…
Comparison theorems on weighted Finsler manifolds and spacetimes with -range
Yufeng Lu, Ettore Minguzzi, Shin-ichi Ohta
We establish the Bonnet-Myers theorem, Laplacian comparison theorem, and Bishop-Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetim…
Quantitative estimates for the Bakry-Ledoux isoperimetric inequality
Cong Hung Mai, Shin-ichi Ohta
We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with . Precisely, we give an upper bound of the volume of the sy…
Geometry of weighted Lorentz-Finsler manifolds I: Singularity theorems
Yufeng Lu, Ettore Minguzzi, Shin-ichi Ohta
We develop the theory of weighted Ricci curvature in a weighted Lorentz-Finsler framework and extend the classical singularity theorems of general relativity. In order to reach thi…
Equality in the logarithmic Sobolev inequality
Shin-ichi Ohta, Asuka Takatsu
We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying . Assuming equality holds, we…
Self-contracted curves in spaces with weak lower curvature bound
Nina Lebedeva, Shin-ichi Ohta, Vladimir Zolotov
We show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound in a sense we introduce in this article. This class of spaces is wide a…
Self-contracted curves in CAT(0)-spaces and their rectifiability
Shin-ichi Ohta
We investigate self-contracted curves, arising as (discrete or continuous time) gradient curves of quasi-convex functions, and their rectifiability (finiteness of the lengths) in E…
Rigidity for the spectral gap on -spaces
Nicola Gigli, Christian Ketterer, Kazumasa Kuwada +1
We consider a rigidity problem for the spectral gap of the Laplacian on an -space (a metric measure space satisfying the Riemannian curvature-dimension condition) fo…
Nonlinear geometric analysis on Finsler manifolds
Shin-ichi Ohta
This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is two-fold: G…
Some functional inequalities on non-reversible Finsler manifolds
Shin-ichi Ohta
We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the appro…
A semigroup approach to Finsler geometry: Bakry-Ledoux's isoperimetric inequality
Shin-ichi Ohta
We develop the celebrated semigroup approach à la Bakry et al on Finsler manifolds, where natural Laplacian and heat semigroup are nonlinear, based on the Bochner-Weitzenböck formu…
Needle decompositions and isoperimetric inequalities in Finsler geometry
Shin-ichi Ohta
Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Levy-Gromov, Bakry-Ledoux, Bayle and E. Milman on weighted Riemannian manifolds. Klartag's…
Gradient flows and a Trotter--Kato formula of semi-convex functions on CAT(1)-spaces
Shin-ichi Ohta, Miklós Pálfia
We generalize the theory of gradient flows of semi-convex functions on CAT(0)-spaces, developed by Mayer and Ambrosio--Gigli--Savaré, to CAT(1)-spaces. The key tool is the so-calle…