Timeline for Do distance functionals separate probability measures?
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| Aug 18, 2020 at 8:48 | comment | added | Christian Bueno | Actually, this idea using antipodal points works for spheres as well. Thus being simply-connected is not a sufficient condition to ensure an affirmative answer either. | |
| Aug 18, 2020 at 8:29 | comment | added | Christian Bueno | Elegant counter-example.This idea can be pushed a bit further as well though it was perhaps implicit: The empirical measure of a pair of antipodal points on the unit circle $\delta_p / 2+\delta_{-p} / 2$ produces an expected intrinsic distance of $\pi/2$ to any point on the circle. Thus, any two "empirical measure of antipodal points" cannot be separated by distance functionals. This shows the obstruction isn't simply due to finiteness of $\Omega$ and that the obstruction can't be avoided by requiring the underlying space to be nonzero dimensional, a manifold, connected, path-connected, etc. | |
| Aug 17, 2020 at 11:13 | history | answered | George Lowther | CC BY-SA 4.0 |