207 research outputs found
Gromov-Hausdorff convergence of discrete transportation metrics
This paper continues the investigation of `Wasserstein-like' transportation
distances for probability measures on discrete sets. We prove that the discrete
transportation metrics on the d-dimensional discrete torus with mesh size 1/N
converge, when , to the standard 2-Wasserstein distance W_2 on the
continuous torus in the sense of Gromov-Hausdorff. This is the first
convergence result for the recently developed discrete transportation metrics.
The result shows the compatibility between these metrics and the
well-established 2-Wasserstein metric.Comment: 22 pages, to appear in SIAM J. Math. Ana
Nilpotency, almost nonnegative curvature and the gradient flow
We show that almost nonnegatively curved m-dimensional manifolds are, up to
finite cover, nilpotent spaces in the sense of homotopy theory and have
C(m)-nilpotent fundamental groups. We also show that up to a finite cover
almost nonnegatively curved manifolds are fiber bundles with simply connected
fibers over nilmanifolds.Comment: minor corrections in the proof of 2.5.1(II
- …