2-Wasserstein (optimal transport) and extension to the set of all signed measures - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T06:34:17Z http://mathoverflow.net/feeds/question/121546 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121546/2-wasserstein-optimal-transport-and-extension-to-the-set-of-all-signed-measures 2-Wasserstein (optimal transport) and extension to the set of all signed measures passerby51 2013-02-12T02:14:20Z 2013-02-12T02:14:20Z <p>Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as $$d_{W_2}(\mu,\nu) = \inf_{\gamma} \int \|x-y\|^2 d\gamma(x,y)$$ where the $\inf$ is over all couplings $\gamma$ of $\mu$ and $\nu$. Can we define a norm (or something norm-like) on the space of signed measures (or a linear subspace of it containing the cone of probability measures) which gives rise to $W_2$ for probability measures. (I suppose not, but why?)</p> <p>If not, can we approximate $d_{W_2}$ by a norm?</p>