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Sharp Rogers--Shephard Inequalities for Rényi Entropy
Bang-Xian Han, Guoqing Zhan, Chiyu Zhou +1
We establish sharp weighted Rogers--Shephard inequalities for Rényi differential entropy. For independent copies of a log-concave random vector in , we prove $h_p(aX-b…
Stability of optimal transport on sub-Riemannian manifolds
Bang-Xian Han, Zhuo-Nan Zhu
We study the stability of optimal transport for the squared Carnot--Carathéodory distance. We prove the sharp stability estimate for optimal transport maps on complete…
Stability of Kantorovich potentials via heat kernel regularization
Bang-Xian Han, Zhuo-Nan Zhu
This survey presents heat kernel regularization as a method for quantitative stability of Kantorovich potentials for the quadratic transport cost. We give a complete new heat kerne…
On the Geometry of Wasserstein Barycenter II: Riemannian Rigidity, Essential Non-Branching, and Finsler Models
Bang-Xian Han, Deng-Yu Liu
Wasserstein barycenters provide a notion of weighted mean for probability measures on a metric space. We prove that the barycenter curvature-dimension condition is…
Sharp Poincaré interpolation along Wasserstein geodesics
Bang-Xian Han, Zhuo-Nan Zhu
Let and be - and -strongly log-concave probability measures on , and let be their quadratic Wasserstein geodesic. We prove the sharp…
Contracting Transport Maps on Riemannian Manifolds
Shrey Aryan, Bang-Xian Han, Zhuo-Nan Zhu
We use inverse mean curvature flow to construct bi-Lipschitz maps that preserve normalized volume and decrease distances. These maps send a round sphere onto any smooth closed stri…
Lipschitz regularity of harmonic map heat flows from spaces into spaces
Bang-Xian Han, Hui-Chun Zhang, Xi-Ping Zhu
We prove positive-time regularity for harmonic map heat flows from finite-dimensional spaces into complete spaces, without assuming that eithe…
Heat Codimension and Scalar Curvature
Bang-Xian Han
Motivated by recent proofs of Gromov's codimension-two volume-growth conjecture, we propose a synthetic theory for scalar lower bound on metric measure Dirichlet spaces.
Weak Quadruple Comparison and Structure Theory Beyond Alexandrov Geometry
Bang-Xian Han, Liming Yin
We introduce a new four-point comparison principle, called the -weak quadruple condition, for non-Riemannian spaces with synthetic non-negative curvature. This con…
Wasserstein Barycenter Convexity Detects Hilbertian Geometry
Bang-Xian Han, Deng-Yu Liu
We prove that convexity of the Boltzmann entropy at Wasserstein barycenters is strong enough to distinguish Hilbert spaces from general Banach spaces. Thus Wasserstein barycenters…
Two-mode stability for multi-marginal optimal transport maps
Bang-Xian Han, Zhuo-Nan Zhu
We establish a two-mode stability theory for Monge solutions of the multi-marginal optimal transport problem with barycentric quadratic cost. The associated tuple of maps splits in…
Stability of optimal transport on metric measure spaces
Bang-Xian Han, Zhuo-Nan Zhu
We prove a quantitative stability of Kantorovich potentials on metric measure spaces with lower Ricci curvature bound, thereby confirming a recent conjecture of Kitagawa, Letrouit…